periodic homogenisatio ofn certai n fully nonlinear
TRANSCRIPT
Proceedings of the Royal Society of Edinburgh 120A 245-265 1992
Periodic homogenisation of certain fully nonlinear partialdifferential equations
Lawrence C EvansDepartment of Mathematics University of California Berkeley CA 94720
USA
(MS received 9 January 1990 Revised MS received 24 April 1991)
SynopsisWe demonstrate how a fairly simple perturbed test function method establishes periodichomogenisation for certain Hamilton-Jacobi and fully nonlinear elliptic partial differential equationsThe idea following Lions Papanicolaou and Varadhan is to introduce (possibly nonsmooth)correctors and to modify appropriately the theory of viscosity solutions so as to eliminate then theeffects of high-frequency oscillations in the coefficients
1 Introduction
We investigate in this paper periodic homogenisation for certain fully nonlinearfirst and second order partial differential equations (PDE) Our basic problem isthis given the PDE
F^D2ueDue uex-^) = 0 (11)
where the nonlinearity F is periodic in its last argument we hope to show that thesolutions ue converge somehow as e goes to zero to a solution u of an effectivelimiting PDE having the form
F(D2u Du u x) = 0 (12)
The primary difficulties are both to discover the precise structure of F and also tojustify rigorously the convergence
We shall narrow our focus to those PDE verifying formally at least amaximum principle so that we may invoke the theory of the weak or so-calledviscosity solutions This approach interprets a solution of ( l l ) e or (12) in termsof its pointwise behaviour with respect to a smooth test function $ (see[76161310] etc)
Such a formulation turns out to be extremely useful for homogenisationquestions where both the bad nonlinearity and the rapid periodic oscillationspreclude any very good uniform estimates on weegto Indeed if we employ thestandard methods of asymptotic expansions to seek a representation
ME = U + poundlaquo + pound2W2+ (13)
Supported in part by NSF grants DMS-86-01532 and DMS-86-10730 (at the Institute for AdvancedStudy Princeton NJ)
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246 Lawrence C Evans
we find formally that both the limit u and the correctors laquo u2 bull bull bull mustnecessarily satisfy various bad nonlinear PDE which in general do not havesmooth solutions On the other hand the earlier paper [9] proposed replacing(13) with a corresponding expansion
0 e = 0 + pound0 + pound202 + (14)
the smooth test function (fgt being given The general hope is that since ltp 1S
regular the appropriate correctors ltplt ltf)2 bull bull bull will be smooth as well This turnsout to be the case for the various quasilinear PDE analysed in [9] For thesituation at hand however the full nonlinearity forces ltpound 02gt bull bull bull and thus ltpe tobe nonsmooth and consequently the techniques of [9] are not applicable But wedemonstrate instead here that nevertheless we can modify viscosity solutiontechniques to handle convergence in various situations in spite of the non-smoothness of our perturbed test functions ltpE In this endeavour we shallemploy a number of ideas from the unpublished paper of Lions Papanicolaouand Varadhan [15] (which predates [9])
In Section 2 we discuss the general theory of homogenisation of certain firstorder fully nonlinear PDE and in Section 3 second order fully nonlinear ellipticequations Section 4 investigates the passage from second order to first orderPDE in the presence of rapid oscillation Finally Section 5 provides severalsimple examples for which more-or-less explicit formulae can be had for theeffective limiting PDE
For more information concerning homogenisation of linear and quasilinearPDE consult [17 2 3] and the references therein
I am very grateful to P-L Lions for many useful suggestions and especially forproviding me with a copy of the unpublished paper [15] several key results ofwhich I reproduce below for the readers convenience
2 First-order equations
To illustrate most clearly the basic techniques let us consider first the modelproblem
r
H(Dueuex-)=0 in Q
( 2 1 )
where Q denotes a bounded smooth open subset of Un and
HWxU x Q x R ^ R
is a given smooth function Our primary hypothesis is that
the mapping ygt-^Hp u x y) is Y-periodic (22)
for all p u x where Y= [0 1] the unit cube in U We additionally require
lim H(p u x y) = +degdeg uniformly on 5(0 L) x Q x U for each Lgt0 (23)lpl-sc
ugt-H(p u x y) mdash fiu is nondecreasing (24)
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Periodic homogenisation of partial differential equations 247
for some pi gt 0 and all p x y and
j H is Lipschitz continuous on
lB(0 L) x 6(0 L)xQxUn for each L gt 0 ^ ^
Suppose finally for each poundgt0 ueeC(Q) is a viscosity solution of (21)e Inview of hypotheses (23) (24) and standard theory for viscosity solutions
sup I K H C ^ Q ^ 0 0 (26)()ltElt1
and thus we may extract a subsequence ueJ=l c MpoundEgt0 and a functionweC01(Q) so that
uegtmdashgtu uniformly on Q as pound--raquobull 0 (27)
We propose to find a nonlinear first order PDE which u solves in the viscositysense The key insight is due to Lions Papanicolaou and Varadhan [15]
LEMMA 21 For each fixed p e IR u e IR and x eQ there exists a unique realnumber A for which the PDE
(H(Dyv+puxy) = k inW
v Y-periodic
has a viscosity solution v e C0gt1(IR)
Proof ([15]) 1 Given p e IR u e IR and x e Ci we consider for each 6 gt 0 theapproximating PDE
dwd + H(Dywd+p u xy) = 0 in IR
Owing to the hypotheses on H there exists a unique Lipschitz viscosity solutionw6 of (29)6 and in particular the uniqueness implies w6 to be y-periodic Wehave in addition the estimate
sup 6wdegL-mpoundH(pux-)L-mltcc (210)0ltlt5ltl
and thus from (23)
(211)UltOlt1
Now set6 (212)
Then v d is F-periodic for each 0 lt 6 lt 1 and
SUp ||tgt6||cltU(y)ltdegdeg (213)0lt6ltl
Utilising (210) (213) we extract a subsequence (v8 (5wlt5)degdeg=1 c
(ua ltW)olt6lt1 so that
-+v uniformly in IR (214)6-gt -A uniformly in IR
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248 Lawrence C Evans
for some function v e C0gt1(IR) v Y-periodic and some constant AeR Passingto limits in the viscosity sense in (29)6 we deduce that the pair (v A) solves(28)
2 Uniqueness of A follows from the comparison theorem for viscositysolutions Indeed suppose (0 X) solves
(H(Dyv+puxy) = i inUn
Iv y-periodic A^
and say X gt A Adding a constant to v if necessary we may suppose as well that
vgtv inU (216)
But for e gt 0 small enough
ev + H(Dyv + puxy)^ev + H(Dyv +puxy) inU
in the viscosity sense whence
v^v in W
a contradiction to (216) bull
To display explicitly the dependence of Aon u and x let us hereafter write
X = H(pux) (peUueUxeQ) (217)
denning thereby the effective Hamiltonian H Thus the cell problem (28) reads
(H(Dyv+puxy) = H(pux) in Wy
v Y-periodic
Following [15] we record next various properties of H inherited from H
LEMMA 22 (a) lim H(p u x) = +ltraquo uniformly on B(0 L) x Q for eachL gt 0
(b) The mapping u gtmdashraquoH(p u x) mdash [iu is nondecreasing for each p x(c) H is Lipschitz continuous on 5(0 L) xB(0L)xQ for each L gt 0(d) If H is convex in p so is H
Various additional properties of H may be found in [15]
Proof 1 Fix Mgt0 Evaluating (29)6 at a point yoe Y where w6 attains itsmaximum we discover
dwd(y0)
Thus hypothesis (23) ensures
provided p is large enough Letting d tend to zero and recalling (214) (217)we find that
H(p ux)^M
if || is sufficiently large This verifies assertion (a)
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Periodic homogenisation of partial differential equations 249
2 Suppose u ipound u dgt0 w6 solves (29)6 and wamp solves
dwd + H(Dywd+puxy) = O inU (219)a
In view of hypothesis (24) and standard comparison principles for viscositysolutions we find
-8wd^-6w6 + fi(u-u) inU
and so upon sending 6 to zero we deduce
H(p u x) s= H(p u x) + fi(u - u)
Assertion (b) is proved3 Choose p eW u u e U x x e Q with p p u u laquo L Let w solve
(29)a and ws solve
p u x y) = 0 in U (220)
Now in view of (25)
H(q +p u x y)^H(q +p u x y) + C(p -p + u - u + x -x)
for all q y with q^L and a constant C = CL L) Thus
dwd + H(Dywd+p u x y)^ -C(p -p + u - u + x -x) in IR
in the viscosity sense This PDE and (29)fi then imply
dwd - dw6 C(p -p + u - u + x - x) in W
and so in the limit
H(p u x) - H(p u x) ^ C(p -p + u-u + x-x)
A similar argument implies as well that
Hp u Jc) - H(p u x) -Cp -p + u-u + x-x)
4 Assume now that H is convex in p Fix p q e IR u e IR x e Q and let vpv9 v(p+q)l2 be Y-periodic viscosity solutions of the PDE
HDyv +p u x y) = H(p u x)q + a u x v = H(a u x in U
(221)H(Dyv + q u x y) = H(q u x) in IR
1 2 ) V 2
Subtracting a constant from v^p+tgtgtn if needs be we may assume
For later contradiction let us suppose that
Y- U x^j gt hHp u x) + iH(q u x) (223)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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246 Lawrence C Evans
we find formally that both the limit u and the correctors laquo u2 bull bull bull mustnecessarily satisfy various bad nonlinear PDE which in general do not havesmooth solutions On the other hand the earlier paper [9] proposed replacing(13) with a corresponding expansion
0 e = 0 + pound0 + pound202 + (14)
the smooth test function (fgt being given The general hope is that since ltp 1S
regular the appropriate correctors ltplt ltf)2 bull bull bull will be smooth as well This turnsout to be the case for the various quasilinear PDE analysed in [9] For thesituation at hand however the full nonlinearity forces ltpound 02gt bull bull bull and thus ltpe tobe nonsmooth and consequently the techniques of [9] are not applicable But wedemonstrate instead here that nevertheless we can modify viscosity solutiontechniques to handle convergence in various situations in spite of the non-smoothness of our perturbed test functions ltpE In this endeavour we shallemploy a number of ideas from the unpublished paper of Lions Papanicolaouand Varadhan [15] (which predates [9])
In Section 2 we discuss the general theory of homogenisation of certain firstorder fully nonlinear PDE and in Section 3 second order fully nonlinear ellipticequations Section 4 investigates the passage from second order to first orderPDE in the presence of rapid oscillation Finally Section 5 provides severalsimple examples for which more-or-less explicit formulae can be had for theeffective limiting PDE
For more information concerning homogenisation of linear and quasilinearPDE consult [17 2 3] and the references therein
I am very grateful to P-L Lions for many useful suggestions and especially forproviding me with a copy of the unpublished paper [15] several key results ofwhich I reproduce below for the readers convenience
2 First-order equations
To illustrate most clearly the basic techniques let us consider first the modelproblem
r
H(Dueuex-)=0 in Q
( 2 1 )
where Q denotes a bounded smooth open subset of Un and
HWxU x Q x R ^ R
is a given smooth function Our primary hypothesis is that
the mapping ygt-^Hp u x y) is Y-periodic (22)
for all p u x where Y= [0 1] the unit cube in U We additionally require
lim H(p u x y) = +degdeg uniformly on 5(0 L) x Q x U for each Lgt0 (23)lpl-sc
ugt-H(p u x y) mdash fiu is nondecreasing (24)
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Periodic homogenisation of partial differential equations 247
for some pi gt 0 and all p x y and
j H is Lipschitz continuous on
lB(0 L) x 6(0 L)xQxUn for each L gt 0 ^ ^
Suppose finally for each poundgt0 ueeC(Q) is a viscosity solution of (21)e Inview of hypotheses (23) (24) and standard theory for viscosity solutions
sup I K H C ^ Q ^ 0 0 (26)()ltElt1
and thus we may extract a subsequence ueJ=l c MpoundEgt0 and a functionweC01(Q) so that
uegtmdashgtu uniformly on Q as pound--raquobull 0 (27)
We propose to find a nonlinear first order PDE which u solves in the viscositysense The key insight is due to Lions Papanicolaou and Varadhan [15]
LEMMA 21 For each fixed p e IR u e IR and x eQ there exists a unique realnumber A for which the PDE
(H(Dyv+puxy) = k inW
v Y-periodic
has a viscosity solution v e C0gt1(IR)
Proof ([15]) 1 Given p e IR u e IR and x e Ci we consider for each 6 gt 0 theapproximating PDE
dwd + H(Dywd+p u xy) = 0 in IR
Owing to the hypotheses on H there exists a unique Lipschitz viscosity solutionw6 of (29)6 and in particular the uniqueness implies w6 to be y-periodic Wehave in addition the estimate
sup 6wdegL-mpoundH(pux-)L-mltcc (210)0ltlt5ltl
and thus from (23)
(211)UltOlt1
Now set6 (212)
Then v d is F-periodic for each 0 lt 6 lt 1 and
SUp ||tgt6||cltU(y)ltdegdeg (213)0lt6ltl
Utilising (210) (213) we extract a subsequence (v8 (5wlt5)degdeg=1 c
(ua ltW)olt6lt1 so that
-+v uniformly in IR (214)6-gt -A uniformly in IR
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248 Lawrence C Evans
for some function v e C0gt1(IR) v Y-periodic and some constant AeR Passingto limits in the viscosity sense in (29)6 we deduce that the pair (v A) solves(28)
2 Uniqueness of A follows from the comparison theorem for viscositysolutions Indeed suppose (0 X) solves
(H(Dyv+puxy) = i inUn
Iv y-periodic A^
and say X gt A Adding a constant to v if necessary we may suppose as well that
vgtv inU (216)
But for e gt 0 small enough
ev + H(Dyv + puxy)^ev + H(Dyv +puxy) inU
in the viscosity sense whence
v^v in W
a contradiction to (216) bull
To display explicitly the dependence of Aon u and x let us hereafter write
X = H(pux) (peUueUxeQ) (217)
denning thereby the effective Hamiltonian H Thus the cell problem (28) reads
(H(Dyv+puxy) = H(pux) in Wy
v Y-periodic
Following [15] we record next various properties of H inherited from H
LEMMA 22 (a) lim H(p u x) = +ltraquo uniformly on B(0 L) x Q for eachL gt 0
(b) The mapping u gtmdashraquoH(p u x) mdash [iu is nondecreasing for each p x(c) H is Lipschitz continuous on 5(0 L) xB(0L)xQ for each L gt 0(d) If H is convex in p so is H
Various additional properties of H may be found in [15]
Proof 1 Fix Mgt0 Evaluating (29)6 at a point yoe Y where w6 attains itsmaximum we discover
dwd(y0)
Thus hypothesis (23) ensures
provided p is large enough Letting d tend to zero and recalling (214) (217)we find that
H(p ux)^M
if || is sufficiently large This verifies assertion (a)
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Periodic homogenisation of partial differential equations 249
2 Suppose u ipound u dgt0 w6 solves (29)6 and wamp solves
dwd + H(Dywd+puxy) = O inU (219)a
In view of hypothesis (24) and standard comparison principles for viscositysolutions we find
-8wd^-6w6 + fi(u-u) inU
and so upon sending 6 to zero we deduce
H(p u x) s= H(p u x) + fi(u - u)
Assertion (b) is proved3 Choose p eW u u e U x x e Q with p p u u laquo L Let w solve
(29)a and ws solve
p u x y) = 0 in U (220)
Now in view of (25)
H(q +p u x y)^H(q +p u x y) + C(p -p + u - u + x -x)
for all q y with q^L and a constant C = CL L) Thus
dwd + H(Dywd+p u x y)^ -C(p -p + u - u + x -x) in IR
in the viscosity sense This PDE and (29)fi then imply
dwd - dw6 C(p -p + u - u + x - x) in W
and so in the limit
H(p u x) - H(p u x) ^ C(p -p + u-u + x-x)
A similar argument implies as well that
Hp u Jc) - H(p u x) -Cp -p + u-u + x-x)
4 Assume now that H is convex in p Fix p q e IR u e IR x e Q and let vpv9 v(p+q)l2 be Y-periodic viscosity solutions of the PDE
HDyv +p u x y) = H(p u x)q + a u x v = H(a u x in U
(221)H(Dyv + q u x y) = H(q u x) in IR
1 2 ) V 2
Subtracting a constant from v^p+tgtgtn if needs be we may assume
For later contradiction let us suppose that
Y- U x^j gt hHp u x) + iH(q u x) (223)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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Periodic homogenisation of partial differential equations 247
for some pi gt 0 and all p x y and
j H is Lipschitz continuous on
lB(0 L) x 6(0 L)xQxUn for each L gt 0 ^ ^
Suppose finally for each poundgt0 ueeC(Q) is a viscosity solution of (21)e Inview of hypotheses (23) (24) and standard theory for viscosity solutions
sup I K H C ^ Q ^ 0 0 (26)()ltElt1
and thus we may extract a subsequence ueJ=l c MpoundEgt0 and a functionweC01(Q) so that
uegtmdashgtu uniformly on Q as pound--raquobull 0 (27)
We propose to find a nonlinear first order PDE which u solves in the viscositysense The key insight is due to Lions Papanicolaou and Varadhan [15]
LEMMA 21 For each fixed p e IR u e IR and x eQ there exists a unique realnumber A for which the PDE
(H(Dyv+puxy) = k inW
v Y-periodic
has a viscosity solution v e C0gt1(IR)
Proof ([15]) 1 Given p e IR u e IR and x e Ci we consider for each 6 gt 0 theapproximating PDE
dwd + H(Dywd+p u xy) = 0 in IR
Owing to the hypotheses on H there exists a unique Lipschitz viscosity solutionw6 of (29)6 and in particular the uniqueness implies w6 to be y-periodic Wehave in addition the estimate
sup 6wdegL-mpoundH(pux-)L-mltcc (210)0ltlt5ltl
and thus from (23)
(211)UltOlt1
Now set6 (212)
Then v d is F-periodic for each 0 lt 6 lt 1 and
SUp ||tgt6||cltU(y)ltdegdeg (213)0lt6ltl
Utilising (210) (213) we extract a subsequence (v8 (5wlt5)degdeg=1 c
(ua ltW)olt6lt1 so that
-+v uniformly in IR (214)6-gt -A uniformly in IR
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248 Lawrence C Evans
for some function v e C0gt1(IR) v Y-periodic and some constant AeR Passingto limits in the viscosity sense in (29)6 we deduce that the pair (v A) solves(28)
2 Uniqueness of A follows from the comparison theorem for viscositysolutions Indeed suppose (0 X) solves
(H(Dyv+puxy) = i inUn
Iv y-periodic A^
and say X gt A Adding a constant to v if necessary we may suppose as well that
vgtv inU (216)
But for e gt 0 small enough
ev + H(Dyv + puxy)^ev + H(Dyv +puxy) inU
in the viscosity sense whence
v^v in W
a contradiction to (216) bull
To display explicitly the dependence of Aon u and x let us hereafter write
X = H(pux) (peUueUxeQ) (217)
denning thereby the effective Hamiltonian H Thus the cell problem (28) reads
(H(Dyv+puxy) = H(pux) in Wy
v Y-periodic
Following [15] we record next various properties of H inherited from H
LEMMA 22 (a) lim H(p u x) = +ltraquo uniformly on B(0 L) x Q for eachL gt 0
(b) The mapping u gtmdashraquoH(p u x) mdash [iu is nondecreasing for each p x(c) H is Lipschitz continuous on 5(0 L) xB(0L)xQ for each L gt 0(d) If H is convex in p so is H
Various additional properties of H may be found in [15]
Proof 1 Fix Mgt0 Evaluating (29)6 at a point yoe Y where w6 attains itsmaximum we discover
dwd(y0)
Thus hypothesis (23) ensures
provided p is large enough Letting d tend to zero and recalling (214) (217)we find that
H(p ux)^M
if || is sufficiently large This verifies assertion (a)
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Periodic homogenisation of partial differential equations 249
2 Suppose u ipound u dgt0 w6 solves (29)6 and wamp solves
dwd + H(Dywd+puxy) = O inU (219)a
In view of hypothesis (24) and standard comparison principles for viscositysolutions we find
-8wd^-6w6 + fi(u-u) inU
and so upon sending 6 to zero we deduce
H(p u x) s= H(p u x) + fi(u - u)
Assertion (b) is proved3 Choose p eW u u e U x x e Q with p p u u laquo L Let w solve
(29)a and ws solve
p u x y) = 0 in U (220)
Now in view of (25)
H(q +p u x y)^H(q +p u x y) + C(p -p + u - u + x -x)
for all q y with q^L and a constant C = CL L) Thus
dwd + H(Dywd+p u x y)^ -C(p -p + u - u + x -x) in IR
in the viscosity sense This PDE and (29)fi then imply
dwd - dw6 C(p -p + u - u + x - x) in W
and so in the limit
H(p u x) - H(p u x) ^ C(p -p + u-u + x-x)
A similar argument implies as well that
Hp u Jc) - H(p u x) -Cp -p + u-u + x-x)
4 Assume now that H is convex in p Fix p q e IR u e IR x e Q and let vpv9 v(p+q)l2 be Y-periodic viscosity solutions of the PDE
HDyv +p u x y) = H(p u x)q + a u x v = H(a u x in U
(221)H(Dyv + q u x y) = H(q u x) in IR
1 2 ) V 2
Subtracting a constant from v^p+tgtgtn if needs be we may assume
For later contradiction let us suppose that
Y- U x^j gt hHp u x) + iH(q u x) (223)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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248 Lawrence C Evans
for some function v e C0gt1(IR) v Y-periodic and some constant AeR Passingto limits in the viscosity sense in (29)6 we deduce that the pair (v A) solves(28)
2 Uniqueness of A follows from the comparison theorem for viscositysolutions Indeed suppose (0 X) solves
(H(Dyv+puxy) = i inUn
Iv y-periodic A^
and say X gt A Adding a constant to v if necessary we may suppose as well that
vgtv inU (216)
But for e gt 0 small enough
ev + H(Dyv + puxy)^ev + H(Dyv +puxy) inU
in the viscosity sense whence
v^v in W
a contradiction to (216) bull
To display explicitly the dependence of Aon u and x let us hereafter write
X = H(pux) (peUueUxeQ) (217)
denning thereby the effective Hamiltonian H Thus the cell problem (28) reads
(H(Dyv+puxy) = H(pux) in Wy
v Y-periodic
Following [15] we record next various properties of H inherited from H
LEMMA 22 (a) lim H(p u x) = +ltraquo uniformly on B(0 L) x Q for eachL gt 0
(b) The mapping u gtmdashraquoH(p u x) mdash [iu is nondecreasing for each p x(c) H is Lipschitz continuous on 5(0 L) xB(0L)xQ for each L gt 0(d) If H is convex in p so is H
Various additional properties of H may be found in [15]
Proof 1 Fix Mgt0 Evaluating (29)6 at a point yoe Y where w6 attains itsmaximum we discover
dwd(y0)
Thus hypothesis (23) ensures
provided p is large enough Letting d tend to zero and recalling (214) (217)we find that
H(p ux)^M
if || is sufficiently large This verifies assertion (a)
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Periodic homogenisation of partial differential equations 249
2 Suppose u ipound u dgt0 w6 solves (29)6 and wamp solves
dwd + H(Dywd+puxy) = O inU (219)a
In view of hypothesis (24) and standard comparison principles for viscositysolutions we find
-8wd^-6w6 + fi(u-u) inU
and so upon sending 6 to zero we deduce
H(p u x) s= H(p u x) + fi(u - u)
Assertion (b) is proved3 Choose p eW u u e U x x e Q with p p u u laquo L Let w solve
(29)a and ws solve
p u x y) = 0 in U (220)
Now in view of (25)
H(q +p u x y)^H(q +p u x y) + C(p -p + u - u + x -x)
for all q y with q^L and a constant C = CL L) Thus
dwd + H(Dywd+p u x y)^ -C(p -p + u - u + x -x) in IR
in the viscosity sense This PDE and (29)fi then imply
dwd - dw6 C(p -p + u - u + x - x) in W
and so in the limit
H(p u x) - H(p u x) ^ C(p -p + u-u + x-x)
A similar argument implies as well that
Hp u Jc) - H(p u x) -Cp -p + u-u + x-x)
4 Assume now that H is convex in p Fix p q e IR u e IR x e Q and let vpv9 v(p+q)l2 be Y-periodic viscosity solutions of the PDE
HDyv +p u x y) = H(p u x)q + a u x v = H(a u x in U
(221)H(Dyv + q u x y) = H(q u x) in IR
1 2 ) V 2
Subtracting a constant from v^p+tgtgtn if needs be we may assume
For later contradiction let us suppose that
Y- U x^j gt hHp u x) + iH(q u x) (223)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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Periodic homogenisation of partial differential equations 249
2 Suppose u ipound u dgt0 w6 solves (29)6 and wamp solves
dwd + H(Dywd+puxy) = O inU (219)a
In view of hypothesis (24) and standard comparison principles for viscositysolutions we find
-8wd^-6w6 + fi(u-u) inU
and so upon sending 6 to zero we deduce
H(p u x) s= H(p u x) + fi(u - u)
Assertion (b) is proved3 Choose p eW u u e U x x e Q with p p u u laquo L Let w solve
(29)a and ws solve
p u x y) = 0 in U (220)
Now in view of (25)
H(q +p u x y)^H(q +p u x y) + C(p -p + u - u + x -x)
for all q y with q^L and a constant C = CL L) Thus
dwd + H(Dywd+p u x y)^ -C(p -p + u - u + x -x) in IR
in the viscosity sense This PDE and (29)fi then imply
dwd - dw6 C(p -p + u - u + x - x) in W
and so in the limit
H(p u x) - H(p u x) ^ C(p -p + u-u + x-x)
A similar argument implies as well that
Hp u Jc) - H(p u x) -Cp -p + u-u + x-x)
4 Assume now that H is convex in p Fix p q e IR u e IR x e Q and let vpv9 v(p+q)l2 be Y-periodic viscosity solutions of the PDE
HDyv +p u x y) = H(p u x)q + a u x v = H(a u x in U
(221)H(Dyv + q u x y) = H(q u x) in IR
1 2 ) V 2
Subtracting a constant from v^p+tgtgtn if needs be we may assume
For later contradiction let us suppose that
Y- U x^j gt hHp u x) + iH(q u x) (223)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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250 Lawrence C Evans
We now claim
in Rraquo (224)
in the viscosity sense To see this set
H
and write
We = r]w
r)e denoting the usual mollifier with support in the ball B(0 E) Then
u xy)^ f)e(y - z)(zgtyw(z) +^~ u x y] dzI JB(yE) V L I
= f r]e(y -z)H(Dyw(z) +^^- u x z) dz + o(l)
r]ey-z)HDyvraquo(z)+puxz)dzZ JB(ye)
+ f le(y - z)H(Dyv(z) + q u x z) dz + o(l)1 JB(ye)
= H(p u x) + H(q u x) + o(l)
as poundmdash0 The last equality is valid since the Lipschitz functions vp and vq solvetheir PDE almost everywhere Sending s to zero we verify (224)
Utilising now (223) (224) we deduce
in
in contradiction to (222) bull
We at last verify that the effective Hamiltonian H determines the limit PDE forour rapidly oscillating problems (21)E
THEOREM 23 Assume H verifies (22)-(25) Then u is the unique viscositysolution of
(H(Duux) = 0 in a
l laquo = 0 ondQ ( 2 - 2 i )
Proof 1 Fix 4gt euro CdegdegQ) and suppose first that u mdash ltp has a strict localmaximum at a point x0 e Q with
laquo(bdquo) = ltp(x0) (226)
We must show that
H(Dlttgt(xo)ltlgt(xo)xo)^0 (227)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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Periodic homogenisation of partial differential equations 251
Let us thus for later contradiction assume to the contrary that
H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)
Now set p = D(f)(xo) u = (jgt(x0) x = x0 in (218) and choose v e C0A(U) to be aviscosity solution of
fH(Dyv + Dltgt(x0) 4gt(x0) x0 y) = H(Dlttgt(x0) 0(xo) x0) = 0 in U
I v y-periodic
Define then the perturbed test function
(^ (xeQ)
Notice that ltpc is Lipschitz continuous but is not C1 in general2 We now claim that
x U 5 ( 4 ^ ^ ^ mB(xor) (230)
in the viscosity sense for some sufficiently small radius r gt 0 to be selected below(Here B(x() r) denotes the open ball with centre xQ radius r) To see this chooseany ip e Cdegdeg(W) and suppose (pe - ty has a minimum at a point x^ e B(x0 r) with
(231)
Then the mapping
has a minimum at x-y and so
ygt-gtv(y)-ri(y)
has a minimum at y =xje for
(232)
Since K is a viscosity solution of (229) we deduce
H(Dr(y) + D(P(x0) ltfgt(x0) x0 yi) ^ 6
Therefore (232) gives
( I ) + Dltp(x0) - DlttgtXl) 4gtx0) Xo j e
Consequently
if r gt 0 is small enough We have verified (230)e
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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252 Lawrence C Evans
3 In view of (21)e (230)e and standard comparison results we conclude
Now send e mdashraquo 0 to find
max3B(xor)
^ max ( u -3B(xor)
a contradiction to the assertion that u mdash (jgt has a strict local maximum at x0 Thus(228) is untenable and so (227) is proved
4 The opposite inequality similarly obtains should u mdash ltfgt have a strict localminimum at a point x0 e Q Consequently u is a (and by uniqueness the)viscosity solution of (225) bull
3 Second-order elliptic equations
We next study periodic homogenisation for fully nonlinear elliptic PDE havingthe form
F[D2us Due ue x - 1 = 0 ir
ue = 0 on dQ
where
is a given smooth function 5x denoting the space of real symmetric n x nmatrices We suppose
the mapping y lt-+F(R p u x y) is F-periodic (32)
for all R p u x and make the additional uniform ellipticity assumption
there exists a constant 6 such that
for all R p u x y Let us hypothesise as well
u gt-gt F(R p u x y) mdash [iu is nondecreasing (34)
for some J gt 0 and all R p x y
F is Lipschitz on Snn x R x R x Q x H (35)
Assume now for each e gt0 that MEeC(Q) is a viscosity solution of (31)EOwing to (33) and estimates of Krylov and Safonov extended to viscositysolutions of elliptic PDE by Trudinger [18] and Caffarelli [5] there exists y gt 0for which
SUp ||Me||cy(Q)lt00- (36)OltElt1
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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Periodic homogenisation of partial differential equations 253
As a consequence we may extract a subsequence laquoe=1 c ueEgt0 and afunction u e COy(Q) with
Megtmdashraquou uniformly on Q (37)
As before we wish to ascertain a nonlinear elliptic PDE which u solves in theviscosity sense
LEMMA 31 For each fixed R e Snx p e W u eU and x e Q there exists aunique real number A for which the PDE
2yv + Rpuxy) = X in Un
v Y-periodic
has a viscosity solution v e CltY(U) for some y gt 0
Proof We mimic the proof of Lemma 21 by considering for lt5gt0 theapproximating problem
6wa + F(D2yw
6 + Rpuxy) = 0 inU (39)a
This PDE has a unique bounded viscosity solution wa (see for instance [1019])which owing to the uniqueness and hypothesis (32) is y-periodic Additionally
sup dW6L-m^F(Rpux-)L~mltlaquogt (310)
0ltlt5ltl
In addition the Krylov-Safonov estimates assert
SUp ||fa||cor(y)ltraquo (311)0lt5ltl
for some y gt 0We now proceed as in the proof of Lemma 21 The C l r regularity follows
from Trudinger [19] bull
We writeX = FRpux) (312)
to exhibit explicitly the dependence of A on R p u x Then (38) reads
(F(D2yv+Rpuxy) = F(Rpux) in W
v y-periodic
LEMMA 32 (a) F is uniformly elliptic in the sense that
F(R+Spux) + 9tr(S)^F(Rpux) ifS^O (314)
for all R p u x(b) The mapping u bull-gt F(R p u x) mdash JXU is nondecreasing for all R p x(c) F is Lipschitz on 5x x Un x U x Q(d) If F is convex in R so is F
Proof 1 Assume 5 = 0 and let vR vR+s be y-periodic viscosity solutions of
(F(DyvR+Rpuxy) = F(RgtPux)
[F(D2yv
R+s + R + Spu x y) = F(R + S p u x) K
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254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
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Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
254 Lawrence C Evans
We may as well suppose additionally
Assume then for later contradiction that
F(R + S p u x) gt F(R p ux)-d tr (5 ) (317)
We now claim that
F(D2yv
R+s + R p u x y)^F(Rp u x) in W (318)
in the viscosity sense To see this let ltfgt e Cdegdeg(U) and suppose vR+s mdash ltp has alocal minimum at a point yoeU In view of (315) we have
F(D2ltjgt(y0) + R + Spu x y0) ^ F(R + S p u x)
Consequently (33) yields
F(D2ltp(y0) + R p u x y0) ^ F(D2(fgtyQ) + R+Spu x0) + 6 tr (5)
^ F(R + S p u x) + 6 tr (S)
gtF(Rpux) by (317)
This establishes (318)Owing now to (315) (318) and comparison theorems for viscosity solutions
we discover
a contradiction to (316) This verifies assertion (a)2 The proofs of assertions (b) and (c) are similar to the corresponding proofs
for Lemma 22 When F is convex in R any viscosity solution of (313) is inCM(R) (and in fact C2r(R) for some y gt 0) The proof of (d) then follows as inLemma 22 bull
THEOREM 33 Assume F verifies (32)-(35) Then u is the unique viscositysolution of
(F(D2uDuux) = 0 in Si
I i i = 0 on da (
Proof 1 The proof is similar to that for Theorem 23 Fix (jgt e Cdegdeg(Q) andsuppose u mdash ltp has a strict local maximum at x0 e Q with
u(x0) = ltKo)- (320)
We intend to prove
FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)
and consequently suppose to the contrary that
F(D2tfgt(x0) Dltpx0) 4gt(x0) xo)=9gt0 (322)
Set R = D2cj)(x0) p = Dltp(x0) u = (px0) x = x0 in (313) and choose v to be a
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
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Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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Periodic homogenisation of partial differential equations 255
viscosity solution of the cell problem2v + D24gt(x0) D4gtxQ) 0 ( o ) x 0 y)
= F(D2(j)(xn) D(j)(x0) ltp(x0) x0) = 9 in R (323)
v F-periodic
Define now the perturbed test function
(X euro Q)
Note that ltpe is C r but is not C2 in general2 We claim that if e gt 0 is small enough then
x) ltpx) gt-^ i n B(xogt r) (324)
in the viscosity sense for some sufficiently small r gt 0
Select ^ pound Ctrade(IR) and suppose ltpE mdash igt has a minimum at a point xx e B(J0gt 0raquowith
(325)
Then the mapping
has a minimum si yx = xjE for
In as much as v is a viscosity solution of (323) we deduce
F(D2ri(yi) + D2cfgt(x0) Dlttgt(x0) lttgt(x0) x0 yi) ^ 6
Thus
( ) Dlttgt(x0) ltKx0) x0 ^ ) ^
whence
F(p2xix) DltKx0) V(i) i ^ ) ^
if r gt 0 is small enough In addition since v e C l y we can compute
Hence
) Iraquo 7 ) = 2
if r e gt 0 are small enough This inequality verifies the claim (324)E
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
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256 Lawrence C Evans
3 Owing now to (31)E (325)e and the comparison theorem for viscositysolutions of fully nonlinear elliptic PDE (cf [1310 12]) we have
Sending e mdashraquo 0 we discover that
^ max (ue-lttgte)3B(xnr)
= max (laquo mdash ltp)3B(xor)
a contradiction since u - ltp has a strict local maximum at x0 This confirms (321)4 The opposite inequality holds similarly provided u mdash ltp has a strict local
minimum at a point xoeQ O
4 Convergence of second-order to first-order equations
As our next general class of problems let us investigate the PDE
F(eD2ue DuE uE x -) = 0 in Q El
= 0 on dQ
forp o n Xn y |T~p i raquogt rrj) v r-v y rr-p n rrp
as above The new effects concern the interplay between the vanishing viscosityterm eD2ue and the high frequency periodic oscillation term xe We shallassume as usual that
the mapping ygt-F(R p u x y) is Y-periodic (42)
for all R p u x Additionally we make the uniform ellipticity assumption (33)We suppose as well that
lim F(0 puxy) = +ltxgt uniformly on B(0 L) X Q X W for each L gt 0 (43)IP -laquobullbullraquo
p u x y) mdash fiu is nondecreasing (44)
for some ju gt 0 and all R p x y Finally we require
F is Lipschitz on SnXn x R x R x Q x R (45)
Now in the light of hypotheses (33) and (44) we have the estimate
sup | |Me| |^Q )^- | |F(000--)IL-(uxY)laquoraquo (4-6)0ltpoundltl fl
In view of the e in front of the second derivatives in (41)E we do not haveavailable Holder estimates as in Section 3 Our plan as before is to show thatthe ue converge uniformly to a viscosity solution u of an effective limit PDE Thenew complication is that we cannot at once legitimately deduce uniformconvergence of any subsequence from the crude bound (46) We instead utilisethe techniques of Ishii [11] and Barles and Perthame [1]
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Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
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Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 257
LEMMA 41 For each fixed p e R u eU and x e Q there exists a unique realnumber A for which the PDE
(F(D2yvDyv+puxy) = X in W
Iv y-periodic
has a viscosity solution v e C1gtV(IR) for some y gt 0
Proof The proof is similar to that for Lemma 31 For each 6 gt 0 we solve
dwd + F(D2wd Dyw8 + p u x y) = 0 in Un (48)fi
and then show
I W6 ~^ uniformly on U (49)
for
- bullY
We display the dependence of A on p u x by writing
X = Fpux) (410)
Equation (47) thus becomes
(F(D2yv Dyv +p u x y) = F(p u x) in U
v y-periodic
LEMMA 42 (a) lim F(p u x) = +degdeg uniformly on B(0 L ) x Q for eachLgt0 lp^degdeg
(b) The mapping u gt-gt F(p u x) mdash fiu is nondecreasing for all p x(c) Fis Lipschitz on WxRxQ(d) F is convex in R and p F is convex in p
Proof 1 Fix M gt0 We consider (48)g and evaluate at a point y 0 e 7 wherew6 attains its maximum Thus
Hypothesis (43) now implies
provided p is large enough Assertion (a) then follows from (49) (410)2 Assertions (b)-(d) are proved as in Lemmas 22 32 bull
LEMMA 43 There exists a unique viscosity solution u e C01(Q) of
(F(Duux) = 0 in QU = 0 on dQ K
Proof This is a consequence of conditions (a)-(c) in Lemma 42 bull
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258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
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264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
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Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
258 Lawrence C Evans
THEOREM 44 Suppose Fsatisfies (33) (42)-(45) Then
uemdashgtu uniformly on Q
where u is the unique viscosity solution of (412)
Proof 1 Set
M(x) = lim sup Mf(z) (x e Q)
u^(x) = iminfuc(z) (xeCl)
ZmdashgtX
Then u is upper semicontinuous M is lower semicontinuous
u^u in Q (413)
Our intention is ultimately to show
u = u = u (414)
u the (unique) viscosity solution of (412)2 We first assert that
F(Duux)^0 inQ (415)
in the viscosity sense To prove this select ltjgt e Cdegdeg(IR) and suppose u - cp has astrict local maximum at x0 e Q with
u(x0) = 0(xo) (416)We wish to show that
F(Dltp(x0) ltHo)o)^0 (417)
and consequently assume to the contrary that
F(Dltfgt(x0) lttgt(x0) xo) = 6gt 0 (418)
Set p = D0(JCO) laquo = 0(o) =o in (411)Let u b e a viscosity solution of (411) and define the perturbed test function
(xeQ)
3 We claim that if e gt 0 is small enough then
)x-^^ in B(x0 r) (419)
in the viscosity sense for some sufficiently small r gt 0 To see this fix ip eand suppose pE mdash p has a minimum at a point jct e fi(x0 r) with
Then
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
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260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
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262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 259
has a minimum at yx = xje for
r)(y)^-pound(xigt(ey)-4gt(ey)) (yeW)
Since v is a viscosity solution of (411) we have
FD2r)(yx) Dr(yi) + Dltp(x0) ltjgt(x0) x0 yx) ^ d
Hence
iD2j)xx) - eD2cj)(x1) Dp(xx) + D(p(x0) - Dcpix^) lttgtxa) x0 zz) = d
and so
F( eD2p(x1) Dip(xx) ty(xx) xlt mdash ) ^ ^ -
provided E rgt0 are small enough This verifies (419)4 Since ue solves (41)e we deduce
max (ue mdash ltgt) = max (ue mdash ltjgte)
B(xor) 3B(xor)
whence we arrive after sending pound -raquo 0 at the contradiction
^ max (u-(jgt)
3B(xor)
5 We next assert
M = 0 on 3Q (420)
Indeed by definition and (41)e we see
u ^ 0 on 3Q (421)To establish the opposite inequality fix any point x0 e dQ and choose a smoothfunction v such that
v(x0) = 0 v gt 0 in Q - xQ Du0onQ (422)Fix a gt 0 so large that
F(OltXDVOX-)^1 (423) el
for each x eQ 0 lt e lt 1 this is possible owing to (43) (422) Then
F[eaD2v aDv ocv x - ) ^ F eaD2v aDv 0 x - ) by (44) (422) el el
^ on Q by (423)
provided e gt0 is small enough By comparison therefore
uE^ av on Q
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
260 Lawrence C Evans
Consequently
M Si av on Q
and thus
laquo(lt) Si 0
As this inequality is valid for each point c0 e dQ we deduce
laquo^0 on dQ
This and (421) establish (420)6 Using now (415) (420) we deduce
wSiu in Q (424)
u the unique viscosity solution of (412) An analogous argument reveals
u SSM in Q
This estimate (424) and (413) lead us at last to (414) bull
5 Some examples and variants
It remains a fascinating and still largely open problem to discover explicitformulae for the effective nonlinearities H and F as above The following aresome partial results in this direction See also Lions Papanicolaou and Varadhan[14]
1 A quadratic Hamiltonian Consider the problem
F(eD2uE DUE ue x -) = (iue - poundaJ-)ueXiX + aJ-)ue
xux - = 0 in Q
lwpound = 0 on3Q
where the smooth symmetric coefficients ay=1 satisfy
y gt-gt atjy) is Y-periodic (1 ^ i j Si n) (52)
andf there exists 6 gt 0 such thatl0|sect|2Sfl(OOsectsecty ( | 6R) (
for all y e W (This example does not verify hypothesis (45) because of thequadratic growth in the gradient but this fact does not affect the followingexplicit analysis) According to Section 4 the corresponding cell problem is
jjuw -aij(y)vyy + aij(y)(vy + pt)vy +) -f(x) = A in Yv y-periodic
Assume temporarily v to be a smooth solution of (54) and set
w = e~v (55)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 261
Then w solves the linear eigenvalue problem
))w i n R lw y-periodic
for
Ww = -aij(y)wyy + 2aij(y)piWy - a^p^jW (57)
As wgt0 we see from (56) and the Krein-Rutman Theorem that fiu mdash (A +()) is the principle eigenvalue Adeg(p) of the operator IT (with periodic boundaryconditions on Y) Hence
Thus as in Section 4
uemdashgtu uniformly on Q
u the unique viscosity solution of
(liu-kdeg(Du)=f inQllaquo = 0 on 3Q
Consider next the first order PDE
H ( D U ue x-) = fiue + aJ-)uexu
ex - f = 0 in Q
El EI
the coefficients ay=i verifying (52) (53) According to Section 2 the cellproblem is
f JIM + a^y)(vyi +pd(vy+Pj) -fx) = A in Uv y-periodic
To solve this let us fix 6 gt 0 and study the approximating problem
u - S2aiJ(y)v6
yiy + a i y laquo ^y-periodic ( 5 - 1 0 ) 6
Setting
we transform (510)fi into the linear eigenvalue problem
wamp v laquoJlaquo (5H)laquo
Since w 6 gt0 we deduce as above that [iu mdash (Aa +f(x))]62 is the principleeigenvalue kdeg(pd) of the operator Lp6 Therefore
f(x) (5gt0) (512)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
262 Lawrence C Evans
It is easy to verify using (53) that
for appropriate constants A B a bgt 0 Thus for fixed p u x
sup |Aa|ltltraquo
In consequence we deduce from (510)6 the bounds
sup ||US||C(Rraquo)lt00
0lt6ltl
provided we add as necessary a constant to vs to ensure
Choose then a subsequence (vdgt Aa)degdeg=1 a (v6 Ad)agt0
uniformly on U
It follows that v and A solve (59) in the viscosity sense In view of the uniquenessof A (Lemma 21) we deduce that in fact
lim A6 = A
Thus the full limit
s 1trade d 2 j Ldeg( |) ( 5 1 3 )
exists for each p e IR We verify also that A is concave and homogeneous ofdegree two Finally we see that the Hamiltonian is
H(px) = -k(p)-f(x)
Theorem 23 now asserts that
ue-u uniformly on Q
u the unique viscosity solution of
(fiu-k(Du)=f in QIM = 0 on 3Q
I do not know any explicit characterisation of the function A(-) beyond therepresentation formula (513)
2 A scalar conservation law with periodic forcing As a rather differentexample we consider next a scalar conservation law driven by a large amplituderapidly oscillating time-periodic forcing term of mean zero The relevant PDE is
u+G(u=-f(x-^j in Rx (0dego)
laquobull = onRxVo) (515)f
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 263
We assume g e LW) n Ldegdeg(U) and also that the smooth function G satisfies
lim G(u) = ltgt (516)
We additionally assume is smooth with
s bull-raquo( s) is 1-periodic (x e U) (517)
I f(xs)ds = 0 (xeU) (518)bullgto
We inquire as usual as to the limit of uE as emdashraquo0 For this set
F(x s) = I f(x t) dt (xseU)
so thats i-gt F(x s) is 1-periodic
andF(x 0) = F(x 1) = 0 (xeR)
Consider the Hamilton-Jacobi PDE
wf + H(wEx-)=0 inRx(0raquo)V el gt gtgt (519)pound
where
h(x) = j g(y)dy (xeU)
andH(p x s) = G(p + F(x s)) (pxseU) (520)
Owing to (516) for each e gt 0 the PDE (519)e possesses a unique viscositysolution wE with the estimates
sup we wl wf||ilaquo(Rx(0raquo))ltooOltpoundlt1
As wEis Lipschitz t
(521)
exists for almost every (x t) In addition uE so defined is the unique entropysolution of (515)e
Pass to a subsequence weJ=1 c weegt0 with
wegt^gtw locally uniformly in U x [0 oo) (522)
for weC01^ x[0 ltraquo)) To find the effective PDE which w solves let usintroduce as in Sections 2-4 an appropriate cell problem which is for the case athand
v + H(Dyv+pxs) = inRx[0raquo)
v 1-periodic in s
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
264 Lawrence C Evans
for p x y e R We solve (523) by seeking a solution v(s) which does not dependon y Thus (523) becomes the ODE
vs + H(pxs) = X inR1gt 1-periodic in s
Setting
u(0) = 0
we compute
v(l) = j us(s) ltfc = A - I H(p x s) ds = 0Jo Jo
provided
A = H(p x)= G(p + F(x s)) ds (p x e R)Jo
As in Section 2 we can show that w is the unique viscosity solution of
(w + H(wxx) = 0 inRx[0co)
w = h o n i x ( = 0 P
Now according to (521) (522)
u x t) -raquo VV(A i) + F(x) = M
weakly-star in Ldegdeg(IR x [0 degdeg)) for
- 1 F(x s) ds = I (pound( 0 dr) ds
In view of (525)
is the unique entropy solution of the conservation law
(ut + H(ux)x = 0 inRx[0oo)laquo = g on R x [0 oo)
Hence u is the unique entropy solution of
(ut + G(u x)x = 0 inRx[0oo)
u=g on R x [0 oo)
where
G(u x) = Hu - F(x) x)=f G(u + F(x s) - F(x)) ds (uxeJo
and
Notice in this example that the initial conditions change in the limitSee [8] for homogenisation of a conservation law with a rapidly oscillating
space-periodic forcing term
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use
Periodic homogenisation of partial differential equations 265
References
1 G Barles and B Perthame Exit time problems in optimal control and the vanishing viscositymethod S1AMJ Control Optim 26 (1988) 1133-1148
2 A Bensoussan L Boccardo and F Murat Homogenization of elliptic equations with principalpart not in divergence form and Hamiltonian with quadratic growth Comm Pure Appl Math 39(1986) 769-805
3 A Bensoussan J L Lions and G Papanicolaou Asymptotic Analysis for Periodic Structures(Amsterdam North-Holland 1978)
4 L Boccardo and F Murat Homogeneisation de problemes quasi-lineaires In Atti del ConvegnoStudio di Problemi-Limite della Analisi Funzionale (September 1981) (Bologna Pitagora 1982)
5 L Caffarelli A note on Harnacks inequality for viscosity solutions of second-order equations(preprint)
6 M G Crandall L C Evans and P L Lions Some properties of viscosity solutions ofHamilton-Jacobi equations Trans Amer Math Soc 282 (1984) 487-502
7 M G Crandall and P L Lions Viscosity solutions of Hamilton-Jacobi equations Trans AmerMath Soc 277 (1983) 1-42
8 Weinan E Homogenization of conservation laws (preprint UCLA 1988)9 L C Evans The perturbed test function method for viscosity solutions of nonlinear PDE Proc
Roy Soc Edinburgh Sect A (to appear)10 H Ishii On uniqueness and existence of viscosity solutions of fully nonlinear second order
elliptic PDEs Comm Pure Appl Math 42 (1989) 15-4511 H Ishii A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations
(preprint)12 H Ishii and P L Lions Viscosity solutions of fully nonlinear elliptic partial differential
equations J Differential Equations (to appear)13 R Jensen The maximum principle for viscosity solutions of fully nonlinear second order partial
differential equations Arch Rational Mech Anal 101 (1988) 1-2714 R Jensen Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential
equations (preprint)15 P L Lions G Papanicolaou and S Varadhan Homogenization of Hamilton-Jacobi equations
(unpublished)16 P L Lions Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations I
Comm Partial Differential Equations 81 (1983) 1101-113417 L Tartar Cours Peccot (College de France February 1977)18 N S Trudinger Comparison principles and pointwise estimates for viscosity solutions of
nonlinear elliptic equations (preprint)19 N S Trudinger On regularity and existence of viscosity solutions of nonlinear second order
elliptic equations (preprint)
(Issued 8 April 1992)
available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use