e.h.a. gonzalez - u. massari variational mean curvatures · 2020. 9. 20. · 2 e.ha. gonzcdez - u....

Rend. Sem. Mat. Univ. Poi. Torino Voi. 52, 1(1994) Partial Diff. Eqs.

E.H.A. Gonzalez - U. Massari

VARIATIONAL MEAN CURVATURES

Abstract. This paper is about the theory of variational meati curvature as introduced in E. Barozzi [BE]. We give also a brief account of the regularity results by De Giorgi-Massari for the sete with variational curvature in Lp(Rn), p > n. Finally we exhibit an example showing that the strong regularity theorem fails in the limit case p = n.

Introductìon

The theory of sets of finite perimeter originated in the work of Ennio De Giorgi in the fifties, on the basis of earlier ideas of Renato Caccioppoli. De Giorgi himself proved the basic compactness and semicontinuity results, which allow the use of the direct methods in the treatment of some classical problems of the Calculus of Variations, such as the Plateau problem and the isoperimetric problem. Many authors successfully àpplied the theory to a number of problems (capillarity, eqùilibrium configurations of fluid masses, surfaces with assigned mean curvature, and so on).

De Giorgi also proved the basic structure theorem for sets with finite perimeter (see Section 1) and the regularity theorem for sets of least area (see Section 3). Roughly speaking, the regularity theorem says that the boundary of a set of least perimeter is an analytic manifold, except possibly for a closed singular set E.

The results of the theory of sets with finite perimeter have greatly contributed to the understanding of first tangential properties of point sets, and one can hope for similar success in dealing with second order differential concepts such as mean curvature. The main subject of this work is to present both some classical and recent results in this direction.

The paper is organized as follows.

In Section 1 we present the basic results we need regarding the theory of perimeters. Sets of finite perimeter are introduced and are shown to have quite reasonable first order properties. For a detailed account we refer to the books [GÈ], [MAM1].

In Section 2 we introduce the concepì of variational mean curvature of a set with finite perimeter. We show that every set E e Rn with finite perimeter has a variational mean curvature HE which is an Z^-runction, we investigate the Lp-integrability of HE and we

2 E.HA. Gonzcdez - U. Massari

prove a basic variational property of HE in £p-spaces. Section 3 is devoted to the classical results of U. Massari about the regularity of sets

E c R " with variational mean curvature HE in Lp(Rn) ,p > n. It is apparent that the regularity theorem fails for p < n, but the limit case p = n was quite elusive and only recently solved, see [GMT]. We prove the (weak) regularity results that hold for p > n and we give a sketch of the proof of the (strong) regularity theorem for p > n.

Finally, in the last section we exhibit an example showing that this last theorem fails for p = n.

1. Sets of Finite Perimeter

A (Lebesgue) measurable set X e Rn is said to have finite perimeter in an open set n c R n if the gradient

\axi oxn )

is a Radon vector measure on n, with finite total variation on Sì. For brevity we denote by

diX the scalar measure -z—$x and by \\dX\\ the total variation of the vector measure D$x-OXi

We then have the formula

(1.1) \\dX\\(A) = supij divG{x)dx : GeC^R"),HGH» < l | ,

that holds for every open set A C Q. Notice that ||dX||(A) = \\d(A - X)\\(A) and that ||aX||- = \\dY\\ whenever

|X — y | + |y — X| = 0, where | • | denotes the Lebesgue measure on Rn. So, in the sequel we shall not distinguish between any two sets differir.g by a nuli set. In particular, the inclusion X cY will mean \X - Y\ = 0.

It is easy to see that the support of \\dX\\ is contained in the boundary dX of X.

Define

(1.2) n(fì) = { X C Q : I^^IKfì) < +00}

I W f l ) = { l c n : \\dX\\{A) < +00 VA CC fì}

REMARK 1.1. Suppose fì n dX is smooth. Then, from the Gauss-Green theorem one easily derives that

n - l , \\dX\\(Q) = H^^QDdX).

Variational mean curvatures 3

Here, as usuai, Hs(s > 0) denotes the s - dimensionai Hausdofff measure in 3Rn, i.e.

(1.3)

H'(X) = ̂ lim inf | ^ ( d i a m X , ) ' : X C U ^ , d i a m X j < 0 >

wa F s'°°(X) = ^ i n f < ^imXjY-.XcUjXj

where^=r(I)S/r(| + l ) .

For seN us is the Lebesgue measure of the unit ball in Rs.

THEOREM 1.1. If two sets X,Y belong to U(Q), so does their intersection and union; moreover, the basic inequality

(1.4) \\d(xuY)\W) + \\d(xnY)m)

4 E.H.A. Gonzalez - U. Massari

Tre2 while

The set E is open, so dE = E - E and tbe Lebesgue measure of dE is

\E-E\>\Q\-\E\>l-?j-,

which is positive for small e > 0. This example shows the existence of open sets with perimeter arbitrarily small but

whose boundary has positive Lebesgue measure.

The reduced boundary

Now, we introduce the notion of reduced boundary d*X of a set XeTl(Q) as the set of those points of SE for which there exists a tangent hyperplane to dE. Denote the derivative of diX with respect to \\dX\\ by vi — VÌ{X\ i.e.

t* f\ i \ i- diX(BJx)) (17) « ( x ) = j%||flW,(»));-Here, as usuai, Bp(a;) = {2/eR

n : ||2/~a;|| < p} denotes the open ball with radious p > 0 centered at xeRn.

Let i/ = ( i / i , . . . ,^ n ) ; from the theorem of Besicovitch on differentiation of measures it follows that v(x) exists and \v{x)\ = 1 for \\dX\\ - almost all' xeQ. Furthermore diX = i>i\\dX\\. The reduced boundary d*X is defined by

(1.8) Sind*X = {xeQndX:\i>(x)\ = l}.

The following theorem shows that we can think of v : n fi d*X —• Sn~1 as a (weak) inner unit normal vector:

THEOREM 1.4. Ut Xell(fì) and xetl n d*X. De/ine

(1.9) Ton(a;) = {yeR" : i/(x) • (y -.a?) > 0} .

Then we have

(1,0) . i m J M ^ l ) ^ ^ \\XnTan(x)nBp(x)\ = u^,

V ' ; p -o+ p n 2 . ' •

1. 2*

REMARK 1.2 Note that (1.11) implies the density Dx(x) at a point a:eftnd*X equals


THEOREM 1.5. (Strutture Theorem) Lei XeU(Q), then

(1.12) Qnd*E = ZU[JCj 3

where ||#A*||(Z) = 0 and each Cj is an hypersurface of class C1. Moreover

(1.13) \\dx\\{(dx-d*x) nfì]=o,

(1.14) \\dX\\(F) = Hn-1(F)VFcd*Xì

(1.15) \\dX\\{A) = Hn-1(And*X)

for every open set A e Q, andfinally

(1.16) nn7Fx = nndX.

REMARK 1.3. From the theory òf differentiation of measures and theorem 1.5 one easily gets the following (weak) version of the Gauss-Green formula:

THEOREM 1.6. (Gauss-Green formula) Let XeII(£2) ; then the formula

(1.17) / dwG{x)dx = - f G*udHn~l

Jnnx Jnnd+x

holds for GeCÌ(Q;Rn).

We conclude this section with the following

THEOREM 1.7. (isoperimetric inequality) Let Xell(£l). Then (1.18) inf{\X\ARn-X\}


REMARK 2.1. It can be easily seen (by computing the first variation of the functional TH) that if E has variational mean curvature H, if H is continuous at xedE, and BE is smooth near x, then the value of the (classical) mean curvature of SE at x is given by -H(x)/^n_1y

For EeU(Rn) define

(2.3) H\E) = {HeLl(*n) ' (2.2)holds}

(2.4) HP{E) = {HeU\E) : HtLp(E)} .

The fact that every set 2?eII(lRn) has a variational mean curvature was observed for the first Urne in [BGT]:

THEOREM 2.1. Let E e Rn be a set with finite perimeter; then

REMARK 2.2. As a matter of fact, to every set E with finite perimeter one can associate a number of curvatures. For, if HeHx(E), then H + Vp > 1, so we really need the hypothesis HP{E) ^

Variational meati curvatures 1

2 let e*. = , - . . keN, k > 1, and define

kl — 1

•' -cfc if lk2

k2 if x2+y2 1.

(b) If the answer to a) is affermative, is it possible to characterize the invoked function minimizing the L-norm? Is that function unique?

(e) Is it possible to give a characterization geometrically meaningful of the function in Theorem 2.2? Does that function depends on p ?

The answers to these questions were given by E. Barozzi in [BE], proceeding as follows:

FIRST: For each e > 0 a curvature He e li1 (E) such that

(2-7) \\He\\Ll(E)


The function HE is defìned by considerìng the minima Ex e E of certain variational functionals B\. Precisely

(2.10) Ex = {x e E : HB{x) > -A} .

The defìnition of HE outside E is similar. So, the answer to question a) is affirmative. Unfortunately, the answer to b) is negative: there are lots of curvatures in ^(E) minimizing the L1(Rn)-norm, see examples below.

THIRD: HE is proved to be the (unique) function invoked in Theorem 2.2.

We shall give now a quick outlook ofBarozzi's construction.

The sets E\ and the function HE.

For the sake of simplicity consider the case \E\ < +oo. Define, for A > 0 and X a measurable subset of E, the functionals

(2.11) BX(X) = \\dX\\ + X\E - X\.

Let E\ be a minimizer of (2.11); from Bx{Ex) < Bx(E) one gets

(2.12) ||^A||+A|£?-£?A| 0 as A —• -hoo. Then, from the semicontinuity of the perimeter (Theorem 1.2 and (2.12)) we readily get

(2.13) lirn \\dEx\\ = \\dE\\ . A-*-foo

It is useful to put (2.12) in the form

(2.14) I g - f t l S 1 * * 1 - 1 ' * 1 . '

Observe that

(2.15) 0 < A < fi = » Ex C E».

So, because \E — U {Ex : A > 0} | = 0, we are able to define

(2.16) HE(x) = -inf {A : xeEx} a.e. x e E.

For x e Rn -E define

(2.17) HE{x) = -HRn_E(x) a.e. x e Rn - £7,

where H^n_E is defìned by (2.16) with £? replaced by Rn - E. Finally, observe that the

Lp(E)-norm of HE may be computed as the Riemann-Stieltjes integrai of the continuous function A —• Ap with respect to the increasing function A —*• \Ex\:

(2.18) / \HE(x)\pdx= f °° \pd\Ex\.

JE JO

Varìational meati curvatures 9

EXAMPLE 2.1. • Let r > 0 and let E be the disk

E = {(x,y) €R2 : x2 + y2


and therefore

d\Ex\ = -rr • c(a), where c(a) = cotg— — A3 v " v ' *2 2 '

Recalling (2.18) we get

J \HE(x,y)\Pdxdy = f ? Y K| + 2V(a) j °° \'-*d\,

so that

HEeHp(E)*=>p


LEMMA 2.1. Let E en(Rn); then

{2m W.èh>w = \\m\ \\HE\\LHW-E) = ll^ll •

Sketch of the proof (for a more detailed proof see [BE]).

Let's prove just the first equality in (2.19), the proof of the secònd one being similar. For simplicity suppose \E\ < +oo and for e > 0 let's choose k € N such that

(2.20) 5f |Bl


In fact, for j = 1 inequality (2.22) is just the inequality Bi(Ei) < Bi(X). Suppose (2.22) holds and prove the similar inequality with'j replaced by j -t-1. From (2.22) with X fi Ej instead of X, from the minimum property of Ej+l and from inequality (1.4) we derive

\\dEJ+1\\ + / Hk(x)dx

= \\dEj+1\\ - \\dEjW + ||d£j| + / Hk(x)dx - i i i | 5 , + 1 |

VH e H\E).

iv) Suppose \E\ < +oo; ifHp(E) ^ for some p, 1 < p < +oo, then HE 6 HP(E) and

(2.24) WHBWLUE) < \W\\LV{E) VH e Hl{E).

file:////dEjW


Moreover, equality tiolds in (2.24) ifandonly if HE(X) = H{x) a.e. x € E.

REMARK 2.6. In this survey we nave sketched the proof of i), ii), and iii). For the proof of iv) see [BE]. Examples (2.1), (2.2) showed that there are many curvatures that minimizes the L1 - norm. In Remark 2.4. it was shown that the condition \E\ < -f-oo can't be avoided in iv).

REMARK 2.7. It can be shown by using a generalized maximum principle (see [MO]) that HE is continuous in Rn - dE. We conclude this section with some other interesting examples.

EXAMPLE 2.3. Let a > 0,S = {{x,y) € R2 : 0 < x < 1, \y\ < xa} , let B be the disk tangent to \y\ = xQ at the points (1,1), (1,-1), and let E = SU B. It is then easy to compute that

0 < a < \ =*> HE e L°°(E);

~ < a < 1 =*• HE e LP(E) if and only if p < n ~ i ; 2 2a — 1

a + 1 a > 1 • =» HE € L

P(£J) if and only if p < a

EXAMPLE 2.4. For the set E e E 2 constructed as in example 2.3 but with xa

replaced by e~^ we easily compute that HE &'l/>(Eytip > 1.

EXAMPLE 2.5. Let {a*} be a sequence of positive numbers decreasing to zero and consider the "tooth-saw" function / : [-ai, ai] —• E defined by / (0) = 0tf(-x) = f(x) and

in the first naif of the interval [afc+i, ak] { x - ak+i dk-X in the second half. Let E = {(#, y) e E2 : - a i < x < au - 1 < y < f{x)}.

Let cfc the middle point of [afc+i, afc], let 7ifc = /(cfc), let Tfc the triangle with vertices at the points (àjg+uO), {ckìhk)ì (akì0) and let Bk the disk with radipus rfc = V^ f̂c centered at (ckì-hk). We nave \Tk - Bk\ = f2 - — J hi and it is not difficult to notice that

HE{x,y)


By choosing the sequence {afe} in such a way that

hk = TTÌÌ—TT2 f o r & > 2, we conclude that k(\\ogk)

HE t LP{E) Vp > 1.

This example shows that even for a set with good regularity properties (note that dE is Lipschitz-continuous) it is possible to have

Hp(E) = V p > l .

In the following example we exhibit a set for which the curvature has a greater integrability but whose boundary has a point with zero density.

EXAMPLE 2.6. Let {c&} a sequence of positive numbers decreasing to zero, let {£?&} a sequence of disjoint disks centered at the points (cfc, 0) with radious r^ > 0. If the ?Vs are small enough, the set E — (jk Bk is globally pseudoconvex, i.e.

We have

and therefore

\\dE\\ < ||


In [SJ] J. Simons was able to prove the non existence of singular minimal cones in Rn for n < 7, and proposed the cone

C = {x e R8 : x\ + x\ + x\ + x\ < x\ + x\ + x27 + x\)

as a possible minimal cone in IR8. The proof of the minimality of C was proved by the first time by E. Bombieri, E. De Giorgi and E. Giusti in [BDG]. Later, a simpler and nice proof of this result was given in [MAM2].

Finally, on the basis of the theorem of Simons, H. Federer [ FH] proved the sharp estimate

Hs(^2)=0 V s > n - 8 . In this Section we present a sketch of the regularity theory for sets with mean curvature

in Hp(Rn) as developed by U. Massari in [MU1],[MU2], thus generalizing the De Giorgi-Federer theorem for sets of least area.

We have organized the exposition of the subject as follows: 1. We introduce the basic tools we need in the following and present some (weak) regularity

results that hold in the case p>n. 2. We define (K, e)-minimal sets and present De Giorgi's lemma. We note that sets in

Hp(Rn), p> rc, are (K, e)-minimal, so De Giorgi's lemma applies.

3. We prove the (strong)-regularity theorem for (K, e)-minimal sets (in Section 4 we present an example showing that this theorem fails in the limit case p = n).

4. Finally, we prove Federer's estimate for the singular set. In the following Q will denote an open subset of IR", Bp(x) = {y eR

n : | y-x \< p}\ E a measurable subset of Q.

LEMMA 3.1. (density) Let E be a set with mean curvature in LP(Q), p > n, let xeQndE. Then /o ,x . ... \EaBJx) I 1 (3.1) min|hmp_0+' | f l ( g | ^-gr ? /««x ... \EnBJx) 1 ^ , 1 (3.2) m a x I K ^ 1 ] ̂ ^ < 1 - - . Let

, . 2 - ( 2 n - l ) 1 / n n ci(n) = na;n ^+ì > °J

then

t**\ mini la ll33IKgp(*))^ - / • . ; . (3.3) min||&mp_>0+ ———i > ci(n) ;

p (3.4) max ||im,_0+ "

nì_? " < -f- • P *

file:///EaBJxfile:///EnBJx

16 E.H.A. Gonzalez - V. Massari

Proof. The result follows from the isoperimetric and Holder inequalities. For a detailed proof see [GMTJ.

Deviation from the minimality

Now we introduce a set funtion $ that yields a measure of how far a set E is from minimizing perimeter in an open set A cQ; precisely, we define

(3.5) ^(E1A) = \\dE\\(A)-inf{\\dX\\(A):{X^E)U(E-X)ccA}.

For A = Bp{x) we write

*(E,Bp(x)) = V(Eìx,p).

The following properties of \& are well known; the proof is an easy consequence of the basic properties of perimeter.

PKOPOSITION 3.1. (èlementary properties of^)

a) monotonicity: A1cA2^ *(£?, Ai) < y(E,A2);

b) semicontinuity: Ej -^ Eoo in L}oc(Q) =^ ty(Eoo,A) < min̂ \\imiif(EjìA). Moreover,

if*(E00,n) = \\imjV(Ej)n)then

(3.6) H^oolKA) = 11^11^11^)

for any open set A c e Q such that \\dEooWidA) = 0.

The following result is proved in [ MM].

LEMMA 3.2. (technical lemma) Suppose BR(0) C Q. Thenfor almost ali ryp, 0 < r < p< R, it holds

f / | 4>B(rx) - E(px) | dH^Hx)]2 < 2 f \y I1"" \\dE\\(y)

(3 7) lA*\ = l J Jr

Varìational meati curvatures 17

A rather important consequence of the estimate (3.9) is obtained for p > n. In fact, for p > n from inequality (3.9) it follows that the function

p*p-nìt{E,x,p)

belongs to L1((0,d)),-d = dist(a:,dQ). Then, recalling (3.7), we derive that the function

(3.10) g{x,p) = pl-^dE\\{Bp{x)) + {n-\)ft-nìt{Eìx,t)dt

Jo is increasing on (0, d) and therefore there exists the limit

(3.11) e(ar)^||tm^af^,rt=J|tm^b+P1"n||^ll(Bp(ir))'

and from (3.3), (3.4) we derive that

(3.12) C l ( n ) < e ( x ) < ^ .

The first inequality in (3.12) can be improved by using (1.10) and (1.16) and the monotonicity of g; one obtains thus

(3.13) u;n_i 0 .

We observe that if E has mean curvature H in Q then XE has mean curvature Hx(x) = X~

lH(x/X) in XQ. This follows from

(3.15). \\dE\\(n) = X1-n\\d(XE)\\{XSl)

and

(3.16) f (j>E(x)H(x)dx = A1 _ n / XE(x)Hx(x)dx.

Jn JXQ Notice that if He LP(Q), p>n, then from (3.16) and Holder inequality we get

(3.17) HxX^?°0 in LJoc(R

n).

Suppose Bt = Bt(0) e XQ; then, recalling (3.4), we derive

(3.18) ||5(AE)Ì|(Bt) = An - 1 | | a E | | ( 5 t / A ) < C 2 ( n ) r -

1 .

From (3.18) and the compactness theorem 1.3 it follows the existence of a sequence Â- -*• +oo such that Ej = XjE -Ì- #«, in L/oc(R

n). From (3.17) we infer that #«, is a set with least area, and from the density estimates (3.1), (3.2) we derive

0

18 EH A. Gonzatez - U. Massari

Suppose p > n\ then, from the monotonicity of the function g introduced in (3.10) we get, recalling (3.11), that

(3.19) Ha^oolK.B,) = lirnll^^lK^e) = lim A^"1 | |^^7||(^/A^ ) = 0(0)*—1 .

Therefore the function t •-• t1~n\\dEoo\\(Bt) is Constant and from (3.7) we obtain

(3.20) Eoo(tx) = Eoo(x) V t > 0

so that Eoo is a minimal cone with vertex 0. Eoo is said the tangent cone to E at the point rr = 0.

REMARK 3.1. The compactness inequality (3.18) fails for p < n, even in the case H e Lp(Rn) V> < n but H & Ln(Rn). To see this, consider the following example for n = 2.

EXAMPLE 3.1. For k e N let

1 1 , 2afc - 1 rk = 7Ò3^> Ofc = 1 — —7=> "fe

and let #fc,J5fc e R2 the disks with radii rk centered at ( ^ + 3 » ^ ) a n d ( ^T3» ~"̂ fc )

respectively. Let Ek the convex hull of Bk U B'k and define E = DkEk. It is not difficult to compute that HE € L

p(R2)Vp < 2. However, for every open set A e R2 such that (0,0) € J4 one easily get

\\d(2mE)\\(A)—+ +oo as m—+ +oo.

REMARK 3.2. Let E e Rn be a set with mean curvature in Lp(Rn), ;> > n, 0 e #£ . Then (see (3.16), (3.17), (3.18)) one easily gets that for every sequence Afe —• oo there exists

k

a subsequence //* such that fik-E —• Eoo, where the set £"00 has zero mean curvature. When n < 7, we have specifically that Eoo is a half-space and from this we conclude that DE(0) = -(vedi [GMT]). For n > 8, this is no more true; for example, for the minimal cone

F = {x e R8 : 4(x? + â + x§) > â + x\ + a^ + x? + xQ

one easily computes that

DF(0) = 1 ^ + iarctg 2) = 0,748971736...

REMARK 3.3. In the case p = n the minimal set Eoo in Remark 3.2 is not unique but depends on the sequence Afe, see the Remark 4.1 below.

{K, e)-minimal sets

Now we introduce the following


DEFINITION 3.1. Let n e Rn an open set, e > 0, K > 0 fixed constants. A set E e ri|OC(fì) is said (K',e)-minimal if the inequality

(3.21) *(Eìxìp) n is (K, e)-minimal with K - un ||#||i,p(n). « = • • P

In the regularity theory a centrai role is played by the so-called excess, defìned by the relation (3.22) u>(E,A) = idE\\(A)~ \ f v(x)\ldEM*) I.

JAndE

defined for open sets A e fi. The following statement holds:

PROPOSITION 3.2.

(a) Ai CA2 C fi =* u(E,Ài) < U(E,AQ) (b) i € f ì f ì d * E =*• limp_0+ />

1_nu;(E,Bp(:c)) = 0.

LEMMA 3.3. (De Giorgi) Let K > 0, e > 0, a € (0,1); tfien there exists a positive Constant

a = a(K, e, a)

such that, for every (K}e)-minimal set E e II/oc(fi), x e fi, v) e (0,


Proof. We made use of the following inequality (see [MAMl]): Let A C B measurable subsets of Q such that ||##||(;4) > 0; then

(3.26)

wàm LE^9Em ~ wim LE^

ì9mx)

Jpy (\\dE\\(B)- \:J v(x)\\dE\\{x) |)] ' . - [\\6E\U)

Let, ioti e (0,/)),

"^ ~ \\dE\\(Bt(x)) JBl(x)naBu{

Vj(x) = i/(x, a-7/?)

x)\\dE\\(x)

and notice that i/j(x) is a Cauchy sequence in Rn„ In fact, from the (K, e)-minimality of E we get

\\dE\\(Balp(x)) > (cSp) n - 1

(3.27) W n - l - — ( ^yV'pr- 1 = ÒIPB-v^-1)

for every j > jo(K, e, a, /»). From (3.26) we infer, for j > jo and ra e N

m —1

I ^+m(a:) - Vj{x) \< J2 \vj+h(x) - fj+h+i(x)\ h=o

(3.28) ^ / n-laU+h)(n-l+e/2)\ V* 1/2 m - 1

< 2 V i -^ - — - 1 = 2 r—5L— 1 a*/4 Y^ OLh€!A

( \ 1/2 i T -77 ) 1 7T« Ì e / 4 =

Vàriational meancurvatures 21

Now let pò = a?°p, t e (0, p0) and let j > jo be such that aj+1p 0 such that

(3.29) Br(x)ndE = Br{x)nd*E

(Le., d*E is open relative to dE). Moreover

(3.30) \v(y)-v{z)\{x)ndE, let fi, m e N. From (3.28) we derive that the inequalities

I "(y) - "h(y) \< c4ahe/4

| ^ ) - ^ + m W I < C 4 a ( / l + w ) £ / 4

hold for h > ho = ho{K,e,p,k). So, inequality (3.30) will follow from the elementary inequality

I »(y) - v{x) \


provided we get an appropriate estimate for | vh(y) - vh+m(z) |. This is done by taking \y - z\< ahpi(l - a m ) , so that BQh+mpi (z) C Bahpi (y) and noticing that in this case

Now choose m e N so that

a < l - a m

and therefore

Plak+i °° and the following.


PROPOSITION 3.3. Let {Ej}, Q, T be as in Lemma 3.4, let A be an open set such that

(3.35) Tf\{dE-d*E)cA.

Then there exists jo = jo{A) such that

(3.36) Tn(djE-d*Ej)cAVj>jo.

Proof. We argue by contradiction. Suppose there exist two sequences jh —• -foo and XH € TD(dEjh - d*Ejh) - A. Without loss of generality we can suppose that the sequence Xh converges to a point x. It is clear that x eT — A. On the other hand, from Theorem 3.1 we derive that

Lj(EjhìBp(xh))>afP-1

holds for Bp(xh) C fì and p at

n~\

Letting t —• p, t < p, we get

u,(EjhìBp(x)>apn-K

Wehave thus

u(E, Bp{x)) = limfc u(EJh, Bp{x)) >apn~y

and therefore (remember Proposition 3.2) x edE — d*E,in contradiction with (3.35).

The following theorem states the link beetween singularities of (K, e)-minimal sets and singular minimal cones.

THEOREM 3.3. Let E be a (Kye)-minimal set in SI e Rn. ìf, for some s>0,

(3.37) Ha[(dE-^E)Qi2]>0.

then there exists a singular minimal cone C e W1 with vertex in x = 0 and such that

(3.38) Hs[(dC-d*C)nBi{0)]>0.


Proof. From well known properties of Hausdorff measures, it follows the existence of a point x0 e (SE -d*E)C\Q such that

\imsup p-3Ha'°°[{dE -d*E)nBp(xo)] > a;s2"5.

P-*O

and therefore there exists a decreasing sequence pj -^ 0 such that

(3.39) Ha>°°[(dE - d*E) n BPi(x0)] > 2"*-W^-

Without loss of generality we can suppose x0 = 0 and the sets Ej = —E converging in Pj

measure to a minimal cone C with vertex at x == 0: From (3.34) and (3.39) we get

Hs>°°[{dC - d*C) n Bi(0)] > UmsupH^KaE^ - d*Ej) h B^O)] 3

= limsup^J^^'^KaE - d*E) n BPi(0)] > 2-s-1u)s > 0

and therefore the invoked statement (3.38).

Federerà estimate on the singular set is now obtained from Theorem 3.3 and the theorem of Simons.

THEOREM 3.4. (Simons [SJ]) Let C e Mn a minimal cone. Then, for n < 7, C is an half-space. In particular dC -d*C — (j>.

THEOREM 3.5. Let E be a (K, t)-minimal set in Q e Mn. Then

(3.40) Hs[{dE -8*E) n-'n] = 0 V à . > n - - 8 .

Proof. We argue by induction on the dimension n. For n = 7 the theorem follows immediatly from theorems 3.3 and 3.4.

Assume that (3.40) holds for a fixed n > 7 and prove it with n replace by n + 1 .

Let E e E n + 1 a (/f,c)-minimal set and, arguig by contradiction, suppose

Hs[(dE -d4E) n n ] > o

for some s > n-\- 1 -& = n — 7. Then, from Theorem 3.3, it follows the existence of a minimal cone C e Mn+1, with vertex at x = .0, such that

(3.41) Hs[{dc -d*c) n#i(o)] > o.

From (3.41) and well known properties of Hausdorff measures we derive the existence of a point xQ e (dC - d*C) n Bx (0), x0 f 0, such that

lim sup praHa>°°l(dC - d*C) n Bp{x0)\ > CJS2~S .

file:///imsup

Vàriational meati curvatums 25

Arguing as in the proof of Theorem 3.3 we get the existence of a minimal cone D C Rn+1, with vertex a?0» such that

Ha(dD-d*D)>0.

Notice that D is in fact a cylinder in the reo-direction. Therefore, the set

A = { a ; 6 R n + 1 : i i i o = 0}

is a minimal cone in Rn such that

Ha-1(d\-d*\)>Oì

which is impossible.

REMARK 3.5. Notice that the only possible singularities for a minimal cone C e R8 is located at its vertex. Therefore, the singularities of a (K, €)-minimal set E e R8 are isolated.

We summarize the results in this section in the following.

THEOREM 3.6. Let£lcRn an open set, let E e n(fi) be a set with meati curvature in LP(fì), OeQndE.

(a) (Weak regularity theorem). Letp > n; thenfor every sequence {Afe} of positive numbers k k m.n

Afe —• +oo there exists a subsequence fik —• +oo and a minimal set Eoo C IR le

depending on the sequence {Xk} such that fikE —• Eoo'in ^?oc(^n)- For the density

DE{®) of E at 0 we have 0 < DE(0) < 1. In the case n < 7 the sets Eoo are

half-spaces and DJS(O) = -•

(b) (Strong regularity theorem). Letp > n; in this case the reduced boundarySlnd*E is a

(n - l)-dimensional manifold of class C1,Q, a > (see Remark 3.4 and theorem 3.2) and Federer's estimate (3.40) holds.

p — n REMARK 3.6. (Some examples and a conjecture) One can ask whether or not - - —

4p is the best Constant in the strong regularity theorem. Let a > 0 and consider the function fa : R -> R defined by

f / , _ j x1+a forrc>0

/ Q W ~ \ 0 forx fa(x)} C R2.

It is not difficult to see that HE e Lp(#i(0)) for p < j — .

With xx~a replaced by -x2 log re one gets a set E e R2 for which

HE e £p(#i(Q)) Wp < +oo and 3E e C1*" Va e (0,1) but dE


We conjecture that the best costant a = a(p) in the strong regularity theorem goes to 1 as p —• +00. Note that even when HE € L°° we cannot expect C1»1 regularity for SE. For example, for

E = {(*, ytz) e R3 : z < (x2 - 2/2)[- log^ 2 + y2)1^2}

we have HE e L°°(Bi) and dE e C1'5 V£ < 1 but dE


the subset of the unit ball # i = 2?i(0,0) lying between the two spirals I\ f. Clearly {(0,0)} = Bi C$dE - d*E), the origin being the only singular point of Bt n dE.

Notice that, for every p e (0,1), the circle x2 -f y2 = p2 has exactly two points lying on dE, namely T(p) and f(p) = -V(p). Moreover, the outward unit normals to SE at T{p) and t(p) coincide; we can then extend v : TUf -4 S1 to a vector field V : Bi - {(0,0)} -• S1

by the obvious requirement

(4.7) V(x,y)=u(x(p)Mp)),x2+y2=P2:

It is now a straightforward computation (see [GMT]) to show that divV e L2(B$. Moreover, from Proposition 4.1 we get that H{x,y) = —divV{x,y) is a curvature for E in # i .

REMARK 4.1. Let rQ be the half-line issuing from 0 with direction (cosa, sena), let tk(a) be the value of the parameter t corresponding to the fc-th intersection of r(t) with ra. One has tk(a) = exp(l - exp(a -f 2&7r)).

By choosing A(fc) = (tfc(a))-1, we see that A(fc) • E converges in Ljoc(R

2) to the half-space through Q with normal vector (- sen a, cos a). Therefore in this case every half-space through 0 is the limit of a suitable sequence of dilations of E.

Aknowledgement. We are grateful to Mrs. Celegato and Mrs. Dalla Costa who did an invaluable typing work.

REFERENCES

[AR] ADAMS R.A., Sobolev Spaces, Academic Press, New York, 1975. [BE] BAROZZI E., The curvature of a boundary with finite area, to appear on Rend. Mat.

Accademia Lincei.

[BGT] BAROZZI E., GONZALEZ E.A.H., TAMANINI I., The mean curvature of a set of finite perimeter, Proc. A.'M.S. 99 (1987), 313-316.

[BDG] BOMBIERI E., D E GIORGI E., GIUSTI E., Minimal cones and the Bernstein problem, Inv. Math. 7 (1969), 243-268.

[DG] D E GIORGIE., Frontiere orientate di misura minima, Sem. Mat. Scuola Nòrm. Sup. Pisa, 1960-1961.

[FH] FEDERER H., The singidar set of area minimizing rectifiable currents with codimension otte and of area minimizing fiat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970), 767-771.

[GÈ] GIUSTI E., Minimal surface andfunctions ofbounded variation, Birkhauser, Boston, Mass., 1984.

[GMT] GONZALEZ E. AH. , MASSARI U., TAMANINI I., Boundaries ofprescribed mean curvature, Rend. Mat. Acc. Lincei, s. 9, 4 (1993), 197-206.

[MU1] MASSARI U., Esistenza e regolarità delle ipersuperfici di curvatura media assegnata in W1, Arch. Rat. Mech. An. 55 (1974), 357-382.

[MU2] MASSARI U., Frontiere orientate di curvatura media assegnata in Lp, Rend. Sem. Mat. Univ. Padova 53 (1975), 37-52.


[MAMl] MASSARI U., MIRANDA M., Minimal Surfaces of Codimension One, North HoUand, Amsterdam, 1984.

[MAM2] MASSARI U., MIRANDA M., A remark on minimal cones, Boll. Un. Mat. Ital. (6) 2-A (1983), 123-125.

[MM] MIRANDA M., Sul minimo dell'integrale del gradiente di una funzione, Ann. Scuola Norm. Sup. Pisa (3) 19 (1965), 627-665.

[MO] MOSCHEN M.P., Principio di massimo forte per le frontiere di misura minima, Ann. Univ. Ferrara 23 (1977), 165-168.

[SJ] SIMONS J., Minimal varieties in riemannian manifolds, Ann. of Math. 2(88) (1968), 62-105.

Eduardo H.A. GONZALEZ Dipartimento di Metodi e Modelli, Università di Padova Via Belzoni 7, 35131 Padova, Italy.

Umberto MASSARI Dipartimento di Matematica, Università di Ferrara Via Machiavelli 35, 44100 Ferrara, Italy.

Lavoro pervenuto in redazione il 18.LI994.

e.h.a. gonzalez - u. massari variational mean curvatures · 2020. 9. 20. · 2 e.ha. gonzcdez - u....

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