periodic homogenisatio ofn certai n fully nonlinear ...rcristof/pdf/wgspring17/evans -...

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)

available at httpswwwcambridgeorgcoreterms httpdxdoiorg101017S0308210500032121Downloaded from httpswwwcambridgeorgcore Carnegie Mellon University on 11 Jan 2017 at 211453 subject to the Cambridge Core terms of use

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)


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



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)êv + 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)



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)



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)



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



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



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) ifSÔ (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



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



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



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]



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



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

uû 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



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



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)



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


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)



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


(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



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



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



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)



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)







sup I K H C ^ Q ^ 0 0 (26)()ltElt1






v Y-periodic






and thus from (23)

(211)UltOlt1

Now set6 (212)











Iv y-periodic A^


vgtv inU (216)




v^v in W






v Y-periodic






dwd(y0)



H(p ux)^M











(29)a and ws solve

p u x y) = 0 in U (220)

Now in view of (25)






and so in the limit







1 2 ) V 2






We now claim

in Rraquo (224)


H

and write

We = r]w






= H(p u x) + H(q u x) + o(l)



in




(H(Duux) = 0 in a

l laquo = 0 ondQ ( 2 - 2 i )



We must show that





H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References


























sup I K H C ^ Q ^ 0 0 (26)()ltElt1






v Y-periodic






and thus from (23)

(211)UltOlt1

Now set6 (212)











Iv y-periodic A^


vgtv inU (216)




v^v in W






v Y-periodic






dwd(y0)



H(p ux)^M











(29)a and ws solve

p u x y) = 0 in U (220)

Now in view of (25)






and so in the limit







1 2 ) V 2






We now claim

in Rraquo (224)


H

and write

We = r]w






= H(p u x) + H(q u x) + o(l)



in




(H(Duux) = 0 in a

l laquo = 0 ondQ ( 2 - 2 i )



We must show that





H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References

























Iv y-periodic A^


vgtv inU (216)




v^v in W






v Y-periodic






dwd(y0)



H(p ux)^M











(29)a and ws solve

p u x y) = 0 in U (220)

Now in view of (25)






and so in the limit







1 2 ) V 2






We now claim

in Rraquo (224)


H

and write

We = r]w






= H(p u x) + H(q u x) + o(l)



in




(H(Duux) = 0 in a

l laquo = 0 ondQ ( 2 - 2 i )



We must show that





H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References





























(29)a and ws solve

p u x y) = 0 in U (220)

Now in view of (25)






and so in the limit







1 2 ) V 2






We now claim

in Rraquo (224)


H

and write

We = r]w






= H(p u x) + H(q u x) + o(l)



in




(H(Duux) = 0 in a

l laquo = 0 ondQ ( 2 - 2 i )



We must show that





H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References






















We now claim

in Rraquo (224)


H

and write

We = r]w






= H(p u x) + H(q u x) + o(l)



in




(H(Duux) = 0 in a

l laquo = 0 ondQ ( 2 - 2 i )



We must show that





H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References























H(Dcj)(x0) 4gt(x0) xo) = 6gt 0 (228)



I v y-periodic


(^ (xeQ)


x U 5 ( 4 ^ ^ ^ mB(xor) (230)


(231)

Then the mapping


ygt-gtv(y)-ri(y)


(232)





Consequently






max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























max3B(xor)

^ max ( u -3B(xor)






ue = 0 on dQ

where


















v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References



























v Y-periodic



6wa + F(D2yw

6 + Rpuxy) = 0 inU (39)a



0ltlt5ltl








v y-periodic






[F(D2yv







We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References

























We now claim that

F(D2yv






^ F(R + S p u x) + 6 tr (S)

gtF(Rpux) by (317)


we discover





I i i = 0 on da (


u(x0) = ltKo)- (320)

We intend to prove

FD24gtxa) D0(o) Hxo) x0) ^ 0 (321)








v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























v F-periodic


(X euro Q)





(325)

Then the mapping




Thus


whence



Hence

) Iraquo 7 ) = 2













= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References































= 0 on dQ
















Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























Iv y-periodic




and then show


for

- bullY


X = Fpux) (410)



v y-periodic














Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References

























Proof 1 Set



ZmdashgtX


u^u in Q (413)


u = u = u (414)


F(Duux)^0 inQ (415)







(xeQ)


)x-^^ in B(x0 r) (419)


Then







Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References


























Hence


and so




B(xor) 3B(xor)


^ max (u-(jgt)

3B(xor)

5 We next assert

M = 0 on 3Q (420)







^ on Q by (423)


uE^ av on Q



Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References






















Consequently

M Si av on Q

and thus

laquo(lt) Si 0


laquo^0 on dQ


wSiu in Q (424)


u SSM in Q






xux - = 0 in Q

lwpound = 0 on3Q







w = e~v (55)





for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























for









ex - f = 0 in Q

El EI




u - S2aiJ(y)v6


Setting




f(x) (5gt0) (512)





sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























sup |Aa|ltltraquo



0lt6ltl



uniformly on U


lim A6 = A

Thus the full limit

s 1trade d 2 j Ldeg( |) ( 5 1 3 )


H(px) = -k(p)-f(x)


ue-u uniformly on Q

















andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References






























andF(x 0) = F(x 1) = 0 (xeR)



where





As wEis Lipschitz t

(521)






v 1-periodic in s





Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References
























Setting

u(0) = 0

we compute


provided




w = h o n i x ( = 0 P





In view of (525)





u=g on R x [0 oo)

where


and





References






















References





















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