Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals

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J. reine angew. Math. 574 (2004), 147—185

Journal fu¨r die reine und angewandte Mathematik ( Walter de Gruyter Berlin
New York 2004

Plane-like minimizers in periodic media: jet flows and Ginzburg-Landau-type functionals By Enrico Valdinoci at Roma

Abstract. This paper is divided into three parts. In the first part, we consider the functional Ð JðuÞ ¼ ai; j ðxÞqi uqj u þ QðxÞwð1; 1Þ ðuÞ dx;

for ai; j and Q periodic under integer translations. We assume that Q is bounded and bounded away from zero, and ai; j is a bounded elliptic matrix. We prove that there exists a universal constant M0 , depending only on n, the bounds on ai; j and Q stated above, such that: given any o A R n , there exists a class A minimizer u for the functional J for which the set fjuj < 1g is contained in the strip fx s:t: x
o A ½0; M0 jojg. Furthermore, such u enjoys the following property of ‘‘quasi-periodicity’’: if o A Q n , then u is periodic (with respect to the identification induced by o); if o A R n  Q n , then u can be approximated uniformly on compact sets by periodic class A minimizers. The functional J introduced above has application in models for fluid jets and capillarity. In the second part of the paper, we extend the previous results to the functional Ð GðuÞ ¼ ai; j ðxÞqi uqj u þ F ðx; uÞ dx;

where ai; j is Lipschitz and satisfies the same assumption mentioned above and F fulfills suitable conditions (roughly, F is a ‘‘double-well potential’’). We prove that, for any y A ½0; 1Þ, there exists a constant M0 , depending only on y; n, the bounds on ai; j and F stated above, such that: given any o A R n , there exists a class A minimizer u for the functional G for which the set fjuj e yg is contained in the strip fx s:t: x
o A ½0; M0 jojg. Also, u enjoys the property of ‘‘quasi-periodicity’’ stated above. In particular, the results apply to the potentials F ðxÞ ¼ QðxÞð1  u 2 Þ, and F ¼ QðxÞj1  u 2 j 2 , which have application in the Ginzburg-Landau model. In the third part, we show how the results above contain, as a limit case, the recent Theorem of Ca¤arelli and de la Llave.

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Valdinoci, Plane-like minimizers in periodic media

Acknowledgments. It is a pleasure to thank Professor Luis Ca¤arelli and Professor Rafael de la Llave: sator arepo tenet opera rotas. I am very indebted to Ovidiu Savin for many deep comments and suggestions. I would like to thank Xavier Cabre´, Ki-ahm Lee, Monica Torres and Yu Yuan for useful discussions, Indam for financial support, y la Universitat Politecnica de Catalunya (Barcelona) e la Scuola Normale Superiore (Pisa) for their warm hospitality during my sojourn there.

Part I. Plane-like minimizers for jets of fluid 1. Introduction. In the first part of the present paper, we consider the functional ð1:1Þ

JW ðuÞ ¼

Ð

W

ai; j ðxÞqi uðxÞqj uðxÞ þ QðxÞwð1; 1Þ ðuÞ dx:

We assume ai; j to be bounded and periodic under integer translations (i.e. ai; j ðx þ eÞ ¼ ai; j ðxÞ for any e A Z n ). We also assume the matrix ai; j to be symmetric and positive definite: l1 e ai; j ðxÞ e L1, with 0 < l e L < þy. We assume that Q is periodic under integer translations, bounded and positive, say 0 < Qmin e Q e Qmax < þy. In two dimensions, the functional above is a model for jets of incompressible, irrotational, inviscid fluids: the interested reader may see, for instance, [BZ] and [Se] for a classical introduction, and also [ACF1] and [ACF3] for recent developments. In particular, the first term in the functional (as shown, for instance, in Lemma 2.4 below) takes into account the Equation of continuity, while the second term (as shown in [AC]) expresses Bernoulli’s law on the free surface. The function u has the physical meaning of a stream function. The particular case ai; j ¼ di; j corresponds to a homogeneous medium. The functional in (1.1) was also dealt with in the theory of minimal surfaces: see, for instance, [Md] and [CC]. The main result of the first part of the present paper is the following: Theorem 1.1. There exists a universal constant M0 , depending only on n, the bounds on ai; j and Q stated above, such that: given any o A R n , there exists a class A minimizer u for the functional J for which the set fjuj < 1g is contained in the strip fx s:t: x
o A ½0; M0 jojg. Furthermore, such u enjoys the following property of ‘‘quasi-periodicity’’:

. If o A Q , then u is periodic (with respect to the identification induced by o). . If o A R  Q , then u can be approximated uniformly on compact sets by periodic n

n

n

class A minimizers.

We say that a minimizer is ‘‘class A’’ minimizer if the functional does not increase under any compact (but not necessarily small) modification (see Definition 7.5 below). The proof of Theorem 1.1 will make use of the results of existence and rigidity of minimizers of [AC], the regularity result of [GG], several density estimates of [CC] and a geometric construction of [CL].

Valdinoci, Plane-like minimizers in periodic media

149

In the second part of this paper, we will extend the results of Theorem 1.1 to more general Ginzburg-Landau-type functionals (see below Section 8). We remark that the results in the second part of the paper also contain the ones of the first part as a particular case. However, we preferred to treat the case of the ‘‘fluid jets’’ separately both because of its physical relevance and because some of the arguments are easier or more explicit in this case. Moreover, the main result of the first part (Theorem 1.1) is slightly stronger than the one proved in the second part (Theorem 8.1). The detailed study of the functional J is also going to show some of the properties of the fluid jets interesting in themselves, as the growth from the free boundary and the regularity issues. The scheme of the first part of the paper is the following: first of all, we consider the ‘‘compact’’ case in which o A Q n . In Section 2, we recall some results of existence and regularity of constrained minimizers. In particular, minimizers are Ho¨lder continuous, with a uniform control on the Ho¨lder constant. We also point out that minimizers are indeed Lipschitz. Also, minimizers satisfy the following rigidity property: if a minimizer takes value y A ð1; 1Þ at some point x, it must take both the values 1 and 1 at some point y, whose distance from x can be uniformly bounded. In Section 3, we recall and derive some density estimates, stating, roughly, that, in some sense, both the sets where a minimizer attains value 1 and where it attains value 1 have ‘‘full dimension’’, while the set where the minimizer is di¤erent from G1 behaves like a set ‘‘of codimension one’’. In Section 4, we discuss the notion of ‘‘minimal minimizer’’. In Section 5, we prove that the minimal minimizer satisfies the so called Birkho¤ property. Roughly, the Birkho¤ property states that the set where the minimal minimizer is in absolute value less than one cannot cross its integer translates. In Section 6, we state and proof an ‘‘abstract’’ Geometric Lemma. The density property will allow us to fit this lemma into our purposes. In Section 7 we prove that, for rational frequencies, the minimal minimizer of strips wide enough is indeed unconstrained and hence class A. Finally, we extend the result to irrational frequencies. Before starting with the technical statements and proofs, we would like to discuss a heuristic (absolutely non-rigorous!) ‘‘proof ’’ of the existence of plane-like minimizers. For simplicity, we will consider now a two-dimensional case; given any vector v A R 2 , let us construct two ‘‘walls’’ in direction v and at distance M from each other. Let us consider a fluid flowing through the ‘‘pipe’’ given by these two walls (see the picture below). This fluid can be thought as being ‘‘constrained’’ by the walls. In some part of the space, the fluid is probably not touching the constraining walls, while, somewhere else, it might stick to them. Let us now consider bigger and bigger M’s, i.e. let us enlarge the pipe. Our ‘‘hope’’ is that, for a suitable (universal!) M, the fluid will not stick to the walls anymore. At this

150

Valdinoci, Plane-like minimizers in periodic media

point, the fluid is actually ‘‘unconstrained’’: we can remove the pipe and obtain a free ‘‘plane-like’’ fluid jet. Obviously, some work needs to be done to make this argument rigorous, especially to show that M is indeed a uniform quantity, independent on the given direction v. This will be precisely the target of the following pages.

A jet of fluid ‘‘constrained between two walls’’.

2. Existence and regularity of minimizers constrained in a strip. In this section, we recall some known properties of existence and regularity of a minimizer for the functional J when we prescribe boundary values for u in a strip. The results of this section are an o¤spring of [AC] and [GG]. In the following, for a vector o A R n and M A R, we denote the strip of size M by o SM :¼ fx A R n s:t: x
o A ½0; M g:

When o A Q n , we can recover compactness by an identification of the space. Namely, we consider the following relation @ induced by o: ð2:1Þ

x @ y i¤

x  y ¼ k A Zn

with o
k ¼ 0:

o o ¼ SM =@ the quotient space induced by such relation (which is topoWe will denote by S~M logically equivalent to the product of an ðn  1Þ-dimensional torus and a real interval).

We also set o o XMo :¼ fu A L 1loc ðS~M Þ Þ s:t: ‘u A L 2 ðS~M

with uðxÞ ¼ 1 for any x s:t: o
x e 0 and uðxÞ ¼ 1 for any x s:t: o
x f Mg: We also denote by L n the n-dimensional Lebesgue measure. In the sequel, we will assume M f 10joj, in order to avoid a degeneracy caused by ‘‘too thin’’ strips.

Valdinoci, Plane-like minimizers in periodic media

151

We start by discussing the existence of such constrained minimizers: Theorem 2.1 (Alt and Ca¤arelli). For any o A Q n and for any M > 0, there exists an absolute minimum of JSMo in the space XMo . Such minimizer is periodic (with respect to the identification @). Proof. The argument of the Existence Theorem 1.3 of [AC] goes through with obvious minor modifications. The assumption o A Q n has been made in order to restrict ourselves to a compact domain, and have that J is finite when evaluated, for instance, on the admissible linear function. r With a little abuse of notation, we sometimes identify the functions in XMo with their o constant extension outside the strip SM . We also denote by MMo the set of the minima found by Theorem 2.1. Comparing JSMo ðuÞ, JSMo ðminfu; 1gÞ and JSMo ðmaxfu; 1gÞ, the reader may easily convince herself that we can assume the minimizers in MMo to take value in ½1; 1. From now on, in order to apply Theorem 2.1, we will restrict our attention to rational frequencies, i.e., if not di¤erently stated, we will assume o A Q n . This restriction will be dropped at the end of Section 7, by an approximation argument. 1 ðR n Þ (see, for It is a standard consequence of Poincare´ inequality that, indeed, u A Hloc instance [S]). Further regularity is addressed by the following

Theorem 2.2 (Giaquinta and Giusti). If u A MMo , then u is C a for any a A ½0; 1Þ. Furo thermore, given any K HH SM , the C a norm of u in K is bounded by a constant depending o only on a; n, the bounds on ai; j and Q and on distðK; qSM Þ. Theorem 2.2 follows, for instance, from [GG] or1) [G2], Theorem 7.6. The interested reader may also see [G], Chapter 5, and, for a di¤erent approach, [DEF]. We now point out that, if the coe‰cients are Lipschitz, the minimizer is actually Lipschitz. Theorem 2.3. Assume that the coe‰cients ai; j are Lipschitz. If u A MMo , then u is o Lipschitz. Furthermore, given any K HH SM , the Lipschitz norm of u in K is bounded by o a constant depending only on n, the bounds on ai; j and Q and on distðK; qSM Þ. Theorem 2.3 can be seen as an extension of the results by Alt, Ca¤arelli and Friedman to the functional in (1.1) under a more general assumption on the coe‰cients. The proof of Theorem 2.3 is deferred to the Appendix, for the reader’s convenience. We would like to remark that Theorem 2.2 is valid even if the coe‰cients ai; j are only bounded, while for the proof of Theorem 2.3 we need the coe‰cients to be Lipschitz. 1) In the introduction of [GG] it is required that the integrand f ðx; z; pÞ depends continuously on ðz; pÞ (although in the actual statement of the theorem in the article this condition is not required, but it is not clear if it is implicitly assumed), which is not the case for the functional we consider here. However, the result in [GG] can be easily made fit to our purpose either by using a limiting argument or by making a direct proof (see for details the author’s dissertation, available on-line at www.math.utexas.edu/mp_arc).

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Valdinoci, Plane-like minimizers in periodic media

The proof of Theorem 1.1 will not use explicitly Theorem 2.3, but only Theorem 2.2. In particular, Theorem 1.1 does not require any regularity assumption on the coe‰cients. We now point out the PDE-properties of the minimizers: Lemma 2.4. Let u A MMo . Then: qi ðai; j qj uÞ ¼ 0 weakly in fjuj < 1g, qi ðai; j qj uÞ f 0 weakly in fu < 1g, and qi ðai; j qj uÞ e 0 weakly in fu > 1g. The proof of this lemma is elementary. 3. Density estimates. We will now recall some density estimates on the level sets of the minimizers. Similar density estimates play a crucial role in several results related to ours: see, for instance [CC] and [CL]. Here, the density estimates will be the tool to apply the Geometric Lemma we will present in Section 6. Several results of this section will be based on [CC]. Theorem 3.1. Assume u A MMo . Then: (i) There exist positive constants c; r0 ; r1 (depending only on n and on the bounds on ai; j and Q) such that JBr ðxÞ ðuÞ e cr n1 ; o . for any r f r0 , provided Brþr1 ðxÞ H SM


(ii) For any y0 A ½0; 1Þ, for any y A ½y0 ; y0  and for any m 0 > 0, if L n BK ðxÞ X fu > yg f m 0 , there exist positive constants c ? ; r0 ; r1 (depending on n, K; m 0 ; y0 and the bounds on ai; j and Q) such that 
L n fu > yg X Br ðxÞ f c ? r n ;


o . Analogously, if L n BK ðxÞ X fu < yg f m 0 , for any r f r0 , provided Brþr1 ðxÞ H SM 
L n fu < yg X Br ðxÞ f c ? r n ;

o . for any r f r0 , provided Brþr1 ðxÞ H SM

We include the proof of this result in the Appendix: our proof is a variation of the one in [CC], where the result was proved in the case ai; j ¼ di; j (we indeed simplify a bit the proof and correct some flawns). From (i) of Theorem 3.1, the following bound on the ‘‘width’’ of fjuj < 1g immediately follows: Lemma 3.2. Let u A MMo . Assume that uðxÞ A ð1; 1Þ. Then, there exist r0 and r1 (depending only on n, the bounds on ai; j and Q) such that u takes either the value 1 or 1 in o Br0 ðxÞ, provided that Br0 þr1 ðxÞ H SM . We would like now to deduce some other density estimates, that will be needed in the application of the Geometric Lemma in Section 6. To do this, we need some better control on the ‘‘width’’ of fjuj < 1g than the one obtained in Lemma 3.2: indeed, in Lemma 3.6

Valdinoci, Plane-like minimizers in periodic media

153

we will show that both the values 1 and 1 are attained in a universal neighborhood of any point in fjuj < 1g. This result will be a consequence of some properties of ‘‘rigidity’’ of minimizers, related to the ‘‘linear growth from the free boundary’’ proved by Alt, Ca¤arelli and Friedman. The following results will lead us to this target. In order to compare the functional in (1.1) with the one dealt with, for instance, in [AC], we observe the following: o Lemma 3.3. Let u A MMo and let W be an open connected set contained in SM , with a Lipschitz boundary. If u does not take the value 1 in W, then u minimizes

Ð

W

ai; j qi vqj v þ QðxÞwfv>1g ;

among all v A L 1loc ðWÞ with ‘v A L 2 ðWÞ and v ¼ u on qW. Analogously, if u does not take the value 1 in W, then u minimizes Ð

W

ai; j qi vqj v þ QðxÞwfv0g ;

among all the functions v A L 1loc ðWÞ such that ‘v A L 2 ðWÞ so that v ¼ u on qW. Assume that J~W ðuÞ < y and that ai; j and Q satisfy the hypotheses stated in Theorem 1.1. Then, for any x A fu > 0g, distðx; fu ¼ 0gÞ e CuðxÞ; provided that Bdistðx; fu¼0gÞ ðxÞ O W, where C is a positive constant, depending only on n and on the bounds on ai; j and Q. Proof. By scaling, we may assume x ¼ 0 and distðx; fu ¼ 0gÞ ¼ 1. Set e :¼ uð0Þ. Since B1 O fu > 0g, the first variation of the functional shows that qi ðai; j qj uÞ ¼ 0 on B1 . Thus, by Harnack inequality, sup u e c  e; B1=2

for a universal constant c  . Let j be the radially symmetric function defined as follows: j ¼ 0 in B1=4 , j ¼ 2c  e

154

Valdinoci, Plane-like minimizers in periodic media

on qB1=2 and growing linearly on the rays of B1=2  B1=4 . Set v :¼ minfu; jg. Then v is an admissible function, so Ð

B1=2

ai; j qi uqj u þ Qwfu>0g e e

Ð

B1=2

ai; j qi uqj u þ

Ð

B1=2

Ð

B1=2

ai; j qi vqj v þ Qwfv>0g

ai; j qi jqj j þ

Ð

B1=2 B1=4

Qwfv>0g :

Since j‘jj e const e and u > 0 on B1=4 , the previous inequality implies e2 f

Ð

B1=2

Qwfu>0g 

Ð

B1=2 B1=4

Qwfv>0g f Qmin jB1=4 j:

r

Lemma 3.5. Let W; u and J~ be as in Lemma 3.4. Let x A qfu > 0g. Then, for any r f 0, sup u f Cr; Br ðxÞ

where C is a positive constant (depending only on n and on the bounds on ai; j and Q), provided Br ðxÞ O W. Proof. By scaling, we may assume x ¼ 0 and r ¼ 1. Since 0 A qfu > 0g, for any e > 0 there exists a point x1 A Be , so that uðx1 Þ > 0. We now construct a polygonal, starting as close as we wish to zero, along which u grows up at least linearly. Set d1 :¼ distðx1 ; qfu > 0gÞ e jx1 j e e: By Lemma 3.4, uðx1 Þ f c  d1 ;

ð3:2Þ

for a suitable constant c  . Let z1 A qBd1 ðx1 Þ X fu ¼ 0g. Define n o r1 :¼ sup r s:t: Br ðz1 Þ O B1 and sup u f c  r : Br ðz1 Þ

By (3.2), r1 f d1 . Also, if Br1 ðz1 Þ touches qB1 , then 1 r1 ¼ 1  jz1 j f 1  d1  e f 1  2e f ; 2 if e e 1=4, so sup u f sup u f B1

Br1 ðz1 Þ

c ; 2

and the claim would be proven. Thus, we may restrict ourselves to the case in which

Valdinoci, Plane-like minimizers in periodic media

155

sup u ¼ c  r1 :

ð3:3Þ

Br1 ðz1 Þ

Take x~1 A Br1 ðz1 Þ attaining the sup in (3.3). In fact, by Maximum Principle, x~1 A qBr1 ðz1 Þ. We want now to show that there exist some universal d > 0 such that ð3:4Þ

x1 Þ; sup u f ð1 þ dÞ uð~

B2r1 ðz1 Þ

provided B2r1 ðz1 Þ O B1 . To prove this, assume x1 Þ sup u < ð1 þ dÞuð~

B2r1 ðz1 Þ

and define vðxÞ :¼

1 uðz1 þ r1 xÞ; r1

Ex A B2 :

Then ð3:5Þ

kvkLy ðB2 Þ e

ð1 þ dÞuð~ x1 Þ ¼ ð1 þ dÞc  ; r1

having used (3.3). Assuming d e 1, we get a uniform bound on kvkLy ðB2 Þ . So, by the Theorem of Giaquinta and Giusti (see [GG] or [G2]), v A C a ðB3=2 Þ and ð3:6Þ

kvkC a ðB3=2 Þ e c~;

for a universal c~. Set w :¼

2v  1; þ dÞ

c  ð1

so that qi ðai; j qj wÞ f 0 in B3=2 and w e 1 because of (3.5). Also, by (3.6) and the fact that wð0Þ ¼ 1, we deduce that L n ðfw e 0g X B3=2 Þ f const. Then, by De Giorgi Oscillation Lemma (see, for instance the Appendix of [C] or Theorem 4.9 in [HL] and references therein), we infer sup w e l, for a universal l A ð0; 1Þ. In particular, since x~1 A Br1 ðz1 Þ, B1

  2 x~1  z1 lfw ¼  1; r1 1þd which is a contradiction if d is small enough. This proves (3.4). By construction, uð~ x1 Þ f uðx1 Þ. Hence, there exists a point x2 A B2r1 ðz1 Þ (and in fact x2 A qB2r1 ðz1 Þ by Maximum Principle) such that uðx2 Þ f uð~ x1 Þ þ dc  r1 f uðx1 Þ þ yjx1  x2 j; for universal d; y > 0, provided that B3r1 ðx1 Þ O B1 .

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Valdinoci, Plane-like minimizers in periodic media

Iterating, we obtain a polygonal of vertices x1 ; x2 ; . . . ; xk such that: for 2 e j e k, xj A qB3rj1 ðxj1 Þ, jxj  xj1 j f rj and ð3:7Þ

uðxj Þ f ð1 þ dÞuðxj1 Þ

ð3:8Þ

f uðxj1 Þ þ yjxj  xj1 j;

provided B3rj1 ðxj1 Þ O B1 . In particular, iterating (3.8), uðxj Þ f ylj where lj is the length of the polygonal joining x1 ; . . . ; xj . We want to prove that we can take j big enough, and make lj bigger than some universal quantity. This would finish the proof of the lemma. Indeed, we must be able to iterate the scheme and find a suitable j  such that jxj  j f 1=20. Otherwise, if jxj j < 1=20 for any j, then rj e jxj  xj1 j < 1=10 for any j; and so B3rj ðxj Þ H B1 for any j, and iterating (3.7) we would get sup u ¼ þy. B1

Therefore, lj  f jxj  j  jx1 j f

1  e. 20

r

We can now sharpen the result of Lemma 3.2: Lemma 3.6. Let u A MMo . Assume that uðxÞ A ð1; 1Þ. Then, there exist r0 and r1 (depending only on n, the bounds on ai; j and Q) such that u takes both the values 1 and 1 in o Br0 ðxÞ, provided that Br0 þr1 ðxÞ H SM . Proof. By Lemma 3.2, u takes, say, the value 1 in Br0 ðxÞ. Without loss of generality, we assume r0 f 3=C, where C is the constant in Lemma 3.5. If u did not assume the value 1 too, using Lemmas 3.3 and 3.5, we get 2 f sup u þ 1 f Cr0 f 3; Br0 ðx0 Þ

which is a contradiction. r We are now in position to improve the previous density estimates: Lemma 3.7. Let u A MMo . Assume that x A fjuj < 1g. Then, there exist positive constants r0 ; r1 and c, depending only on n and on the bounds on ai; j and Q, such that JBr ðxÞ ðuÞ f cr n1 ; o . for any r f r0 , provided Brþr1 ðxÞ H SM

Proof. By Lemma 3.6, Theorem 2.2 and Theorem 3.1    9 f const r n L Br ðxÞ X u > 10 n

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Valdinoci, Plane-like minimizers in periodic media

and    9 f const r n : L n Br ðxÞ X u <  10 Let a be the average of u in Br ðxÞ. If a e 0,    9 n 9 ðu  aÞ f L Br ðxÞ X u > ju  aj f 10 10 Br ðxÞXfu>9=10g Br ðxÞ Ð

Ð

f const r n :

Analogously, if a f 0,    9 n 9 f const r n : ju  aj f L Br ðxÞ X u <  10 10 Br ðxÞ Ð

So, in any case, using Poincare´ and Cauchy Inequalities: const r n1 e

Ð 1 Ð ju  aj e const j‘uj r Br ðxÞ Br ðxÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 
Ð j‘uj 2 L n Br ðxÞ X fjuj < 1g e const Br ðxÞ

e const

 Ð

Br ðxÞ

ai; j qi uqj u þ Qwð1; 1Þ ðuÞ :

r

Let us summarize some of the density estimates obtained in this section: Proposition 3.8. Let u A MMo . Assume that x A fjuj < 1g. Then, there exist positive constants r0 ; r1 ; c and C, depending only on n and on the bounds on ai; j and Q, such that: JBr ðxÞ ðuÞ f cr n1 ;  

L n Br ðxÞ X fu > 1g f cr n and L n Br ðxÞ X fu < 1g f cr n ;

o o . Moreover, for any y A SM , for any r f r0 , provided Brþr1 ðxÞ H SM

JBr ð yÞ e Cr n1 ; o . for any r f r0 , provided Brþr1 ðyÞ H SM

4. The minimal minimizer. Following the ideas developed in [CL], we now turn our attention to a special minimizer of J, namely the minimal minimizer. The minimal minimizer is defined by o uM ðxÞ :¼ inf o uðxÞ: u A MM

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Valdinoci, Plane-like minimizers in periodic media

By definition, for any u A MMo and y A R, o > yg O fu > yg: fuM

ð4:1Þ

o We now want to prove that uM is still a minimizer: this will be accomplished in the following Proposition 4.3. For this, we need some easy auxiliary observation. o o Þ, Þ with ‘u; ‘v A L 2 ðS~M Lemma 4.1. For any u; v A L 1loc ðS~M

JSMo ðuÞ þ JSMo ðvÞ f JSMo ðminfu; vgÞ þ JSMo ðmaxfu; vgÞ: The proof is a straightforward computation. Corollary 4.2. If u; v A MMo , then minfu; vg; maxfu; vg A MMo . Proof. Since JSMo ðuÞ e JSMo ðminfu; vgÞ and JSMo ðvÞ e JSMo ðmaxfu; vgÞ, by Lemma 4.1 we deduce JSMo ðuÞ ¼ JSMo ðminfu; vgÞ and JSMo ðvÞ ¼ JSMo ðmaxfu; vgÞ. r o A MMo . Proposition 4.3. uM

The proof of Proposition 4.3 is now a standard application of the Theorem of Ascoli and it is omitted. 5. Birkho¤ property for the minimal minimizer. In this section, we prove a geometric property about the global behavior of the minimal minimizer. This property can be seen as a generalization of the so called Birkho¤ property in Aubry-Mather theory (see [M], [Ma] and [MF]). A similar property has been pointed out in Proposition 8.1 of [CL] in the minimal surface setting. If k A R n , we denote by Tk the translation by the vector k. Definition 5.1. Let E O R n , $ A R n . We say that E satisfies the Birkho¤ property with respect to the vector $ if:

. For any k A Z . For any k A Z

n

so that k
$ e 0, Tk E O E.

n

so that k
$ f 0, Tk E P E.

Proposition 5.2. Let E O R n . Assume that E satisfies the Birkho¤ property with respect to o. Then, there exists a constant %, depending only on n, such that: if CðEÞ contains a ball of radius %, then CðEÞ contains a strip of width 1 that intersects the ball. The proof is elementary. o Proposition 5.3. Let y A R. Then, the set fuM > yg satisfies the Birkho¤ property with respect to o. Namely:

. If k A Z . If k A Z

n

o o > yg. > yg P fTk uM and k
o e 0, then fuM

n

o o > yg. > yg O fTk uM and k
o f 0, then fuM

Valdinoci, Plane-like minimizers in periodic media

159

o o Proof. Set v :¼ minfuM ; Tk uM g. If k
o e 0, by Corollary 4.2, v is a minimizer in o o o the strip Tk SM . By construction, Tk uM is the minimal minimizer in Tk SM . Thus, from (4.1), o o > yg: > yg O fv > yg O fuM fTk uM o Analogously, if k
o f 0, v is a minimizer in SM , so o o > yg O fv > yg O fTk uM g: fuM

r

Using the Birkho¤ property, it is easy to see that the minimal minimizer enjoys the so called doubling property (also known in the literature as no-symmetry-breaking-property), i.e.: the minimal minimizer of period multiple of the original one agrees with the original one. 6. The Geometric Lemma. This section is devoted to the statement and the proof of a geometric lemma. The application of this ‘‘abstract’’ lemma to our case will be given in Proposition 6.3. Lemma 6.1 (Geometric Lemma). Let b H E O R n , with qE O qb, m and n be Radon measures on R n . Take x0 A b. Fix r > 0. Assume that there exist d 0 A ð0; 1=10 such that: (i) Ex A Br ðx0 Þ and Er A ½d 0 r; r=10, 
m Br ðxÞ e const r n :

(ii) Ex A b X Br ðx0 Þ and Er A ½d 0 r; r=10


 n Br ðxÞ f const r n1 :


(iii) m E X Br ðx0 Þ f const r n .


(iv) n B2r ðx0 Þ e const r n1 .

Then, there exists d, depending only on n and on the constants involved in (i)–(iv) above, so that, if d 0 e d, we can find x A Br ðx0 Þ such that ð6:1Þ

Bdr ðxÞ H ðE  bÞ X B2r ðx0 Þ:

A similar lemma can be stated when assumption (ii) above holds when b is a dense subset of qE (for instance, the ‘‘reduced boundary’’, as in the case of the minimal surfaces: see [CL] and, for instance, [G1] or [MM] for the theory of minimal surfaces). In the case studied in [CL], m is the Lebesgue measure and n is the Haussdor¤ measure on the reo duced boundary. In our application, m will be the Lebesgue measure and nðBÞ :¼ JB ðuM Þ. Namely, we will apply the Geometric Lemma to the sets E :¼ fu > 1g, or E :¼ fu < 1g, and b :¼ fjuj < 1g. The hypotheses of the Geometric Lemma will be satisfied using the density estimates of Section 3. The rough idea of the proof of the Geometric Lemma is the following. We first cover

160

Valdinoci, Plane-like minimizers in periodic media

b with balls of radius dr: since b behaves like an ðn  1Þ-dimensional set, we need Oð1=d n1 Þ of such balls to cover b. We then extend this cover to the whole set E: since E behaves like an n-dimensional set, we need Oð1=d n Þ of such balls to cover E. Therefore, if d is small enough, there is at least one ball in the cover of E that does not touch b.

The Geometric Lemma.

In order to make the argument above rigorous, we need the following auxiliary covering argument: Lemma 6.2. Assume the hypotheses (ii) and (iv) of Lemma 6.1. Fix d A ½5d 0 ; 1=2. Then, there exists a family of points yk A b, k ¼ 1; . . . ; N, such that:

. B ðy Þ X B ðy Þ ¼ j if k 3 h. . b X B ðx Þ O S B ðy Þ. . N e const d . k

dr=5

dr=5

h

N

r

0

dr

k

k¼1

ðn1Þ

Proof. Take any y1 A b X Br ðx0 Þ. If b X Br ðx0 Þ O Bdr ðy1 Þ, we can stop. Otherwise, take y2 A b X Br ðx0 Þ not belonging to B2dr=5 ðy1 Þ. Iterating this procedure, assume to have a family of yk ’s such that yk A b X Br ðx0 Þ for 1 e k e N, verifying Bdr=5 ðyk Þ X Bdr=5 ðyh Þ ¼ j N S if k 3 h. If b X Br ðx0 Þ O Bdr ðyk Þ, we stop; otherwise select yNþ1 A b X Br ðx0 Þ outside k¼1 N S B2dr=5 ðyk Þ. k¼1

We now estimate the ‘‘good’’ N for which we stop the procedure: in light of hypothesis (ii) of Lemma 6.1,

Valdinoci, Plane-like minimizers in periodic media

n

S N

k¼1

161

P N
 n Bdr=5 ðyk Þ f const NðdrÞ n1 : Bdr=5 ðyk Þ ¼ k¼1

Also, by construction, Bdr=5 ðyk Þ O B2r ðx0 Þ. Hence, hypothesis (iv) of Lemma 6.1 implies that S N n Bdr=5 ðyk Þ e const r n1 : k¼1

These two inequalities show that N e const d 1n .

r

We now address the proof of Lemma 6.1: Proof. Consider a finite overlapping cover B of Br ðx0 Þ, consisting in balls of radius dr, with KB e const dn . Since B is finite overlapping, any of the balls fBdr ðyk Þg1ekeN , constructed in Lemma 6.2 cannot intersect more than C~ balls belonging to B, where C~ is a universal constant.

since

Then, out of the collection B, only const dðn1Þ (at most) may intersect b X Br ðx0 Þ, ð6:2Þ

KfB A B s:t: B X b X Br ðx0 Þ 3 jg eKfB A B s:t: B X Bdr ðyk Þ 3 j for some kg e C~N e const dðn1Þ :

We now prove that, if d is chosen suitably small, at least one ball B  of the collection B must be contained in E  b. If not,  S  S S B ; B W B¼ BAB

BAB BOCðEÞ

BAB BXb3j

so, by (6.2) and hypotheses (i) and (iii),   S 
 B const r e m E X Br ðx0 Þ e m E X n

BAB

    S   S B þm EX B em EX BAB BOCðEÞ

BAB BXb3j

  S  B ¼0þm EX BAB BXb3j

e

P

BAB BXb3j

mðBÞ e const dðn1Þ ðdrÞ n ;

which is a contradiction if d is small enough. Such a ball B  satisfies the desired property (6.1). r

162

Valdinoci, Plane-like minimizers in periodic media

Following is the application of the Geometric Lemma to our case: Proposition 6.3. Let u A MMo . There exist positive constants k; r0 and r1 , depending only on n and on the bounds on ai; j and Q, such that: for any x A fjuj < 1g and any r f r0 , there exist z1 and z2 , such that Bkr ðz1 Þ O fu ¼ 1g X Br ðxÞ;

and

Bkr ðz2 Þ O fu ¼ 1g X Br ðxÞ;

o . provided B2rþr1 ðxÞ H SM

Proof. Apply the Geometric Lemma to the sets E :¼ fu > 1g, or E :¼ fu < 1g, o and b :¼ fjuj < 1g. Take m :¼ L n and nðBÞ :¼ JB ðuM Þ. The hypotheses of the Geometric Lemma are fulfilled in light of Proposition 3.8. r 7. The minimal minimizer is unconstrained and class A. Proposition 7.1. There exists M0 > 0, depending only on n and on the bounds on ai; j o j < 1g does not touch the upper constraint and Q, such that if M f M0 joj, then the set fjuM o o fo
x ¼ Mg. Also, Ea f 0, uM ¼ uMþa . Proof. Let S be a strip of width 3 around the hyperplane L :¼ fx
o ¼ M=2g. First of all, we will prove that there exists a strip H of width 1 parallel to the constraints and at distance universally bounded from L, on which o o ¼ 1: ¼ 1 or uM either uM

ð7:1Þ

For that, we consider two cases: o Case 1: there exists a point x  A S so that uM ðx  Þ A ð1; 1Þ. Then, from Proposition 6.3 there exists a ball (whose radius is big if M is big), contained in o o fuM ¼ 1g ¼ CfuM > 1g. By the results in Section 5, we obtain a strip of width 1 intero secting such a ball and contained in fuM ¼ 1g. o o ðSÞ O ðy; 1 W ½1; þyÞ. By the continuity of uM , Case 2: uM o either uM ðSÞ O ðy; 1

o or uM ðSÞ O ½1; þyÞ.

In any case, (7.1) is proved. With no loss of generality, assume also o1 > 0. Now we o o infer that it is not possible that uM ¼ 1g satisfies the jH ¼ 1: Otherwise, since the set fuM Birkho¤ property with respect to o (see Section 5), we obtain that S S o o Tðj; 0;...; 0Þ fuM ¼ 1g O fuM ¼ 1g; Tðj; 0;...; 0Þ H O jAN

jAN

o thus uM ¼ 1 in the whole portion of space trapped between the lower constraint and H. But o then we can consider the integer translation v :¼ Tð1; 0;...; 0Þ uM , which has the same energy o o as uM , contradicting the fact that uM is the minimal minimizer. o o Therefore, uM jH ¼ 1. Then, again, since the set fuM > 1g satisfies the Birkho¤ property with respect to o, for any j A N,

Valdinoci, Plane-like minimizers in periodic media

163

o > 1g ¼ j; Tð j; 0;...; 0Þ H X fuM o so uM ¼ 1 in the whole portion of space trapped between the upper constraint and H. r

In order to clearly state the next result, we introduce the following notation. If a > 0, we define n o SM; a :¼ fx A R s:t: x
o A ½a; M þ ag

and 2 ~o 1 o ~o XM; a :¼ fu A L loc ðSM Þ s:t: ‘u A L ðSM Þ

with uðxÞ ¼ 1 for any x s:t: o
x ¼ a and uðxÞ ¼ 1 for any x s:t: o
x ¼ M þ ag: Theorem 7.2. There exists a constant M0 , depending only on n, the bounds on ai; j and o Q, so that if M f M0 joj, then the minimal minimizer uM is an unconstrained minimizer, i.e. o o for any a > 0 and any u A XM; a , JSM; o ðu Þ e J o ðuÞ. S M a M; a Proof. Proposition 7.1 above establishes that the upper constraint fx
o ¼ Mg is irrelevant. Hence, we have only to show that the lower constraint L 0 :¼ fx
o ¼ 0g is not necessary either. Given a > 0, take k A Z n so that k
o > a: for instance, if o1 3 0, take k :¼ A
signðo1 Þe1 with A A N su‰ciently big. And consider translations Tk u. r Remark 7.3. Following [CL], we would like to sketch an alternative geometric argument for proving that the minimal minimizer is unconstrained. Let % be the constant in Proposition 5.2. We assume, without loss of generality, that %, M=4 A N and that all the components of o are multiple of 4%. Let us consider a non-overlapping covering of the o strip SM made by cubes. If K o is the number of cubes of side 4% intersecting the plane fo
x ¼ 0g, we have that the number of cubes in the covering is MK o =4. On the other hand, let n be the number of cubes in the covering whose dialation of a factor 1=2 intero sects the set fjuM j < 1g. We want to prove that n < MK o =4: from this, we would obtain o the existence of a cube of side 2% outside fjuM j < 1g. In the same way as above, this and the Birkho¤ property would imply that the minimizer is unconstrained. To show that n < MK o =4, we first notice that, comparing with a one-directional o linear function interpolating 1 and 1, one sees that JSMo ðuM Þ e const K o . And, by Lemma o 3.7, if a cube Q of size 2% intersects fjuj < 1g, then J2Q ðuM Þ f const. From these two estimates, it follows that n e const K o and so n < MK o =4, provided that M is big enough. A similar geometric construction has been used in [CL] for dealing with manifolds with negative curvature. Remark 7.4. From Theorem 7.2 and Theorem 2.2, it follows that, if M f M0 joj, o then, for any a A ½0; 1Þ, kuM kC a ðR n Þ e C ? , for a suitable C ? , depending only on a; n and on o the bounds on ai; j and Q. More precisely, by Theorem 2.3, kuM kC 0; 1 ðR n Þ e C ? .

164

Valdinoci, Plane-like minimizers in periodic media

Definition 7.5. We say that u is a class A minimizer for the functional J if for any admissible function v, verifying v ¼ u in CðKÞ, we have JK ðuÞ e JK ðvÞ. Notice that the definition of class A requires the functional not to increase under any compact (but not necessarily small) modification. This property was introduced for geodesics in [Mo], and it appeared also in Aubry-Mather theory. Theorem 7.6. There exists a constant M0 , depending only on n, the bounds on ai; j and o Q, so that if M f M0 joj, then the minimal minimizer uM is a class A minimizer. o as periodic with Proof. Given a compact perturbation, we can always consider uM a period larger than the diameter of the perturbation and (in light of Theorem 7.2) as a minimizer in a strip of width bigger than the size of the perturbation. r

In order to prove Theorem 1.1, it su‰ces now to consider the case of an irrational frequency o A R n  Q n , since the rational frequency case was dealt with in the previous pages. An irrational frequency o will now be handled passing to the limit rational frequencies on approximating o. on . By Remark 7.4 and the Theorem of AsNamely, consider minimal minimizers uM on coli, uM converges to some uy uniformly on compact sets. Accordingly, uy is a class A minimizer. This completes the proof of Theorem 1.1.

Part II. Plane-like minimizers for Ginzburg-Landau-type models 8. The Ginzburg-Landau-type functional. In the second part of this paper, we will deal with the functional GW ðuÞ ¼

ð8:1Þ

Ð

W

ai; j ðxÞqi uðxÞqj uðxÞ þ F ðx; uÞ dx:

The assumptions on the coe‰cients ai; j are analogous to the one in Part I. Moreover, we assume in this part that the coe‰cients ai; j are Lipschitz (indeed, the constants in the density estimates may depend, in this case, on the Lipschitz norm of the coe‰cients); F is supposed to be non-negative and bounded, periodic under integer translations of the xvariable, and F ðx; 1Þ ¼ F ðx; 1Þ ¼ 0. We also assume that inf F ðx; tÞ f gðyÞ, where gðyÞ jtjey

is a decreasing, strictly positive function in the interval ½0; 1Þ, and that there exist d A ½0; 2 and l A ð0; 1Þ so that:

. F ðx; tÞ f constð1  jtjÞ , if jtj A ðl; 1Þ. . F is continuous in u, for juj < 1; F is defined and in L . If d > 0, F ðx; uÞ is continuous for juj < 1, and d

u

y loc

for any juj < 1 and a.e. x.

u

Fu ðx; 1 þ sÞ f const s d1 ; if s < l.

Fu ðx; 1  sÞ f const s d1 ;

Valdinoci, Plane-like minimizers in periodic media

165

. If d ¼ 2, F is continuous in u for juj e 1, F ðx; uÞ is increasing for u near 1 and u

increasing for u near 1.

Important examples are given by the potentials F ðxÞ ¼ QðxÞð1  u 2 Þ and F ¼ QðxÞj1  u 2 j 2 , with Q positive and periodic: indeed, functionals similar to the latter were introduced in [GL] for the study of phase transition problems occurring in superconductivity and later applied in superfluid models (see [GP]). For a discussion about phase transition models, see, for instance, [LL], p. 206. For a related ‘‘gradient penalization method’’ in the theory of phase transitions, see the discussion in [Gu], p. 188. See also [R], which is a translation of van der Waals’ pioneering work2) and presents an interesting historical review. The main result of the second part of the present paper is the following: Theorem 8.1. There exists a universal constant M0 , depending only on n, the bounds on ai; j and F stated above, such that: given any o A R n , there exists a class A minimizer u for the functional G for which the set fjuj e 99=100g is contained in the strip fx s:t: x
o A ½0; M0 jojg. Furthermore, such u enjoys the following property of ‘‘quasi-periodicity’’:

. If o A Q , then u is periodic (with respect to the identification induced by o). . If o A R  Q , then u can be approximated uniformly on compact sets by periodic n

n

n

class A minimizers.

The reader may compare Theorems 8.1 and 1.1: notice, in particular, that in Theorem 1.1 it was possible to control the set fjuj < 1g in a uniform way and no regularity of the coe‰cients was required there. The proof of Theorem 8.1 will use some of the machinery already developed in the first part of the paper. The main obstacle in shadowing the proof presented above consists in the fact that we do not expect in general to have the rigidity and non-degeneracy properties stated in Lemmas 3.4 and 3.5. Namely, for smooth potentials, the minimizers do not reach the values 1 and 1, so, in this case, we are not in presence of a free boundary. However, the arguments involving the minimal minimizer and the Birkho¤ property still hold. Therefore, to extend the previous proof we have to obtain the density estimates that fit our purposes by another argument. The proof will go as follows: In Section 9 we discuss the existence and regularity of constrained minimizers. The arguments will be in analogy with the ones in Section 2; the main di¤erence is that, in this case, we can not exactly prescribe the values on the boundary of the strip to be G1: this will be replaced by the weaker constraint of requiring the function to be bigger than, say, 99/100 if o
x e 0 and smaller than 99/100 if o
x f M. Obviously, the explicit choice of the values G99=100 is by no means important. In Section 10 we discuss the density estimates that we need. From this, it follows that the minimal minimizer is unconstrained and class A, provided that the strip is wide enough. 2) The Dutch original appeared in Verhand. Konink. Akad. Wetensch. Amsterdam (Sect. 1) 1 (1892), no. 8, 1–56; Jbuch 24 (1967).

166

Valdinoci, Plane-like minimizers in periodic media

9. Existence and regularity of minimizers constrained in a strip. As in the first part, we first turn our attention to a rational frequency o. The irrational frequency case will then be handled by a limit argument. We recall now the equivalence relation ‘‘@’’ defined in (2.1), and we define ~ n :¼ R n =@ (topologically, R ~ n is the product between an ðn  1Þ-dimensional torus and a R real line). We consider the space of admissible functions 1 ~n ðR Þ YMo :¼ fu A Hloc

with uðxÞ f 99=100 for any x s:t: o
x e 0 and uðxÞ e 99=100 for any x s:t: o
x f Mg: Following is the extension of the Existence Theorem 2.1 we need: Theorem 9.1. For any o A Q n and for any M > 0, there exists an absolute minimum in YMo of the functional GR~ n ðuÞ :¼

Ð

~n R

ai; j qi uqj u þ F :

Such minimizer is periodic (with respect to the identification @). 2o
x þ 1 in fo
x A ½0; M g, and continuously extend u 0 as a M ~ n . For any R > 0, define DR :¼ fo
x A ½R; Rg and D ~R :¼ DR =@ (topologconstant to R ~ ically, DR is the product between an ðn  1Þ-dimensional torus and an interval). Proof. Set u 0 :¼ 

Consider a minimizing sequence uk : by cutting o¤ the values above 1 and below 1, we may assume 1 e uk e 1. We may also assume GR~ n ðuk Þ e GR~ n ðu 0 Þ, hence k‘uk kL 2 ðR~ n Þ is bounded. From these facts and Poincare´ inequality, it follows that kuk kL 2 ðD~R Þ is bounded. ~R Þ (and a.e.) and Therefore, we may assume that uk converges to a suitable u in L 2 ðD 2 ~ that ‘uk converges to ‘u weakly in L ðDR Þ, so that lim inf k!y

Ð

ai; j qi uk qj uk f lim inf k!y

~n R

Ð

ai; j qi uk qj uk f

~R D

Ð

ai; j qi uqj u;

~R D

and, by Fatou’s Lemma and the assumptions on F , lim inf k!y

Ð

~R D

F ðx; uk Þ f

Ð

~R D

F ðx; uÞ:

r

The set of minimizers obtained via Theorem 9.1 will be denoted by MMo ðGÞ. The minimizers whose existence has just been proven are C a , for a suitable a A ½0; 1Þ, with a uniform bound on the C a norm in the interior: this follows from Giusti and Giaquinta’s Theorem (see [GG]). Also, in the case of smooth coe‰cients ai; j and a smooth potential F , further regularity can be obtained by applying standard elliptic estimates to the Euler-Lagrange

Valdinoci, Plane-like minimizers in periodic media

167

equation of the minimizers. Indeed, in this case, by inspecting the first variational of the functional, it is easy to see that minimizers verify 2qi ðai; j qj uÞ ¼ Fu : 10. Density estimates. Theorem 10.1. Assume u A MMo ðGÞ. Then: (i) There exist positive constants c; r0 ; r1 (depending only on n and on the bounds on ai; j and F ) such that GBr ðxÞ ðuÞ e cr n1 ; o . for any r f r0 , provided Brþr1 ðxÞ H SM


(ii) For any y0 A ½0; 1Þ, for any y A ½y0 ; y0 ? and for any m 0 > 0, if L BK ðxÞ X fu > yg f m 0 , there exist positive constants c ; r0 ; r1 (depending on n; K; m 0 ; y0 and the bounds on ai; j and F ) such that 
L n fu > yg X Br ðxÞ f c ? r n ; 
o for any r f r0 , provided Brþr1 ðxÞ H SM . Analogously, if L n BK ðxÞ X fu < yg f m 0 , 
L n fu < yg X Br ðxÞ f c ? r n ; n

o for any r f r0 , provided Brþr1 ðxÞ H SM .

The proof of this result, which is deferred to the Appendix, is a modification of the one in [CC], where the result was proved in the case ai; j ¼ di; j . We now derive some easy consequences from the previous result: Corollary 10.2. Fix y0 A ½0; 1Þ. Assume u A MMo ðGÞ and uðxÞ A ½y0 ; y0 . Then, there exist positive constants c; r0 ; r1 (depending only on y0 ; n and on the bounds on ai; j and F ) such that  

L n fu > y0 g X Br ðxÞ f cr n and L n fu < y0 g X Br ðxÞ f cr n ;

o for any r f r0 , provided Brþr1 ðxÞ H SM .

Proof. Define y  :¼ ð1 þ y0 Þ=2. From the fact that u is Ho¨lder continuous and (ii) of Theorem 10.1,  

L n fu > y  g X Br ðxÞ f cr n and L n fu < y  g X Br ðxÞ f cr n ; ð10:1Þ

so that

 

L n fu > y  g X Br ðxÞ  L n fu > y0 g X Br ðxÞ 
e L n fy  < u e y0 g X Br ðxÞ e

Ð 1 F;  gðy Þ fy  y  g X Br ðxÞ  L n fu > y0 g X Br ðxÞ e cr n1 :

From this and (10.1),


L n fu > y0 g X Br ðxÞ f const r n  const r n1 f const r n ;

if r big enough. r

In the sequel, following a standard notation in Geometric Measure Theory (see, for instance [G1]), we denote the perimeter of a Borel set E in an open set W as o nÐ PðE; WÞ :¼ sup div j s:t: j A C01 ðW; R n Þ; jjðxÞj e 1 : E

The following result completes the density estimates we need in the second part of the paper: Lemma 10.3. Fix y0 A ½0; 1Þ. Assume u A MMo ðGÞ and uðxÞ A ½y0 ; y0 . Then, there exist positive constants c; r0 ; r1 (depending only on y0 ; n and on the bounds on ai; j and F ) such that 
L n fjuj < y0 g X Br ðxÞ f cr n1 ;

o . for any r f r0 , provided Brþr1 ðxÞ H SM

Proof. Define 8 y0 g or E :¼ fu < y0 g and b :¼ fjuj < y0 g; also, the measure m and n appearing in the ‘‘abstract’’ Geometric Lemma will be taken here to be the Lebesgue measure and the restriction of the Lebesgue measure to the set fjuj < y0 g, respectively. In the following Proposition 10.6, we will apply Proposition 10.5 with y0 ¼ 99=100, to get uniform bounds.

170

Valdinoci, Plane-like minimizers in periodic media

Proposition 10.6. There exists M0 > 0, depending only on n and on the bounds on ai; j o and F, such that if M f M0 joj, then the set fuM > 99=100g does not touch the upper cono o straint fo
x ¼ Mg. Also, Ea f 0, uM ¼ uMþa . The proof of Proposition 10.6 is analogous to the one of Proposition 7.1. The rest of the proof of Theorem 8.1 follows exactly as in Section 7.

Part III. Back to the minimal surfaces 11. The result of [CL] as a limit case. Now we point out that the results of [CL] about plane-like minimal surfaces can be recovered as a limiting case of jet flows or Ginzburg-Landau minimizers. Namely, we now turn our attention to the e-dependent functional GWe ðuÞ :¼

ð11:2Þ

1 eai; j qi uqj u þ F ðuÞ dx: e W Ð

The assumptions on ai; j and F will be the same as in Section 8 and e > 0 is a small parameter3). It is known (see [Md], [B], [OS] and [CC]) that the minimizers ue of (11.2) converge (when e ! 0, up to subsequences) to a step function, and that the level sets of ue converge Ð pffiffiffiffiffiffiffiffiffiffiffiffiffi uniformly to an interface, which minimizes ai; j ni nj dH n1 . Here, as customary, q ? U q ?U

denotes the reduced boundary of a Caccioppoli set U, and n is its normal. This has the following geometric interpretation (see [BPV] and the author’s thesis for details): if we consider a Riemannian metric g on R n , with a suitable choice of ai; j , minimizers of Ð pffiffiffiffiffiffiffiffiffiffiffiffiffi ai; j ni nj dH n1 are simply minimal surfaces with respect to the notion of perimeter q ?U

induced by g. Also, the non-degeneracy of g translates into the uniform ellipticity of ai; j and, if g is periodic, the coe‰cients ai; j are periodic as well. Thus, the result we now address is the following:

Theorem 11.1. There exists a universal constant M0 , depending only on n, the bounds on ai; j and F, but independent of e, such that: Given any o A R n , there exists a class A minimizer u e for the functional G e , for which the set fju e j e 99=100g is contained in the strip fx s:t: x
o A ½0; M0 jojg. Furthermore, such u e enjoys the following property of ‘‘quasi-periodicity’’:

. If o A Q , then u is periodic (with respect to the identification induced by o). n

e

3) For simplicity, we take here F independent of x: a more general situation can be dealt with using the same techniques, following [B].

171

Valdinoci, Plane-like minimizers in periodic media

. Fixed e > 0, if o A R  Q , then u can be approximated uniformly on compact sets n

n

e

e

by periodic class A minimizers of G .

In the case of the jet flows, the result is actually uniform in y and no regularity of the coe‰cients is needed. In particular, sending e to zero, we re-obtain the main result4) of [CL]: Theorem 11.2. Let g be a bounded metric on T n . There exists a universal constant M (depending only on n and on the nondegeneracy of g), such that, given any o A R n , there exists a codimension 1 surface contained in the strip fo A ½0; Mjojg, which is class A minimal with respect to the perimeter induced by g. Also, if o is rational, the surface obtained is periodic (with respect to the identification induced by o); if o is irrational, the surface obtained can be approximated locally uniformly by periodic class A minimal surfaces. Following a notation introduced in the previous sections, we will denote by MMo ðG e Þ o the set of minimizers among the admissible functions in the strip SM : see page 4 and 19 for details. The proof of Theorem 11.1 readily follows from the work done in the first and second part of the paper, using the following rescaling: for any function u we define R e uðxÞ :¼ uðexÞ:

ð11:3Þ

Notice that the renormalization R e preserves the double-well potential and the ellipticity constants, but does not preserve integer periodicity. Therefore we can not apply the arguments involving the Birkho¤ property to R e u. Thus, we only use the rescaling for extending the Geometric Lemma to R e u, since this result did not use the integer periodicity. Then, apply the Birkho¤ property arguments directly to u. Remark 11.3. A natural question arising in this context is whether the minimal surfaces that are plane-like minimizers are always graphs. The reader may easily convince herself that the answer to such question, in general, is no, just looking at a periodic metric on R 2 which is very small on a set with an ‘‘S’’ shape and very big elsewhere. Analogously, one can construct examples of metrics whose plane-like minimizers for the functional (11.2) are not monotone in any direction.

Appendix 12. Proof of Theorem 2.3. The proof will be accomplished in several steps. Lemma 12.1. Fix qi ðai; j qj vÞ e 0, and

m > 0.

Let

b A ð0; 1Þ.

Assume

that

v A H 1 ðBr Þ,

4) Theorem 11.2 was also proved by [Mo] and [H] in the case n ¼ 2 and by [Ba] in the case n ¼ 3.

v f 0,

172

Valdinoci, Plane-like minimizers in periodic media

vfm

on qBr X fxn e ð1 þ bÞrg:

Then,
there exists a constant y A ð0; bÞ, depending only on n; b and the bounds on ai; j , such  that: v 0; . . . ; ð1 þ yÞr f mb=2.

Proof. By scaling, it is enough to prove the claim for r ¼ 1. Set c ? :¼ 2nkai; j kW 1; y . Let c be the solution of qi ðai; j qj cÞ ¼ c ? m in B1 , with c ¼ 0 on qB1 . By elliptic estimates (see, for instance [GT], Theorem 8.16 and Theorem 8.27) ð12:4Þ


 c 0; . . . ; 0; ð1 þ yÞ e oscB XB 1

2y ð0;...; 0; 1Þ

c e cy a m;

for suitable constants c and a, depending only on n and the bounds on ai; j . Define ð12:5Þ

f :¼ m½ðxn þ 1Þ þ b

and z :¼ f þ c:

We have qi ðai; j qj zÞ ¼ qi ai; n m þ c ? m > 0: We now check that v f z on qB1 . Indeed, if x A qB1 X fxn e 1 þ bg, zðxÞ ¼ fðxÞ e mb e vðxÞ; on the other hand, if x A qB1 X fxn > 1 þ bg, zðxÞ e 0 e vðxÞ. Hence, by Maximum Principle, v f z in B1 . In particular, by (12.4) and (12.5), vð0; . . . ; 0; 1 þ yÞ f mðb  cy a  yÞ f mb=2, if y is small enough with respect to b. r We would like to remark that an alternative proof of the lemma above can be accomplished by using the boundary Harnack inequality. o verifies Lemma 12.2. Let u A MMo . Suppose that x0 A SM o Þ > 2 distðx0 ; fjuj ¼ 1gÞ: distðx0 ; qSM

Then, minf1  uðx0 Þ; 1 þ uðx0 Þg e C distðx0 ; fjuj ¼ 1gÞ; where C is a suitable constant, depending only on n, the bounds on ai; j and F. Proof. If uðx0 Þ ¼ G1, d 0 :¼ distðx0 ; fjuj ¼ 1gÞ.

there

is nothing

to prove. Set

UG ¼ 1 G u,

and

We assume that minfUþ ðx0 Þ; U ðx0 Þg f Md 0 , and derive an upper bound for M. Notice that B9d 0 =10 ðx0 Þ H fjuj < 1g, so by Lemma 2.4 and Harnack inequality ð12:6Þ

inf

B3d 0 =4ðx0 Þ

UG f const
sup UG f const Md 0 : B3d 0 =4ðx0 Þ

Let y A Bd 0 ðx0 Þ X fjuj ¼ 1g. Without loss of generality, we may assume y ¼ 0. We now denote, for short, B :¼ Bd 0 and U :¼ UG.

Valdinoci, Plane-like minimizers in periodic media

173

Let v :¼ vG be the minimum of Ð

ai; j qi zqj z

B

among the functions z A H 1 ðBÞ with z ¼ UG on qB. Hence, ð12:7Þ

Ð Ð const j‘ðU  vÞj 2 e ai; j qi ðU  vÞqj ðU  vÞ B

B

¼

Ð

B

Ð ai; j qi uqj u  ai; j qi vqj v: B

By construction, V ¼ VG :¼ Gðv  1Þ ¼ u on qB: hence by (12.7) and the fact that u A MMo , ð12:8Þ

Ð

B

Ð j‘ðU  vÞ 2 j e const Qwð1; 1Þ ðV Þ  wð1; 1Þ ðuÞ: B

By Maximum Principle and the fact that 0 e U e 2, we deduce that either 0 < v < 2 or v is constant on B. In the latter case, since u A MMo , u must be constant in B: in particular uðx0 Þ ¼ uð0Þ ¼ G1, and the claim is proven. Thus, we can now restrict ourselves to the case 0 < v < 2 in B, i.e. 1 < V < 1. From (12.8), Ð

ð12:9Þ

B

j‘ðU  vÞj 2 e const L n ðfjuj ¼ 1g X BÞ:

From (12.6) and Lemma 12.1, there is a suitable constant k A ð0; 1Þ and a suitable point x ? A Bkd 0 so that vðx ? Þ f const Md 0 . Harnack inequality then implies ð12:10Þ Set now v  ðxÞ :¼

inf v f const Md 0 :

Bkd 0

1 vðd 0 xÞ, for x A B1 . Let us now introduce the following barrier: d0 2

wðxÞ ¼ cMðemjxj  em Þ; with c and m suitable constants, properly chosen to satisfy qi ðai; j qj wÞ > 0 in B1  Bk ; w ¼ 0 on qB1

and

w e v  on qBk ;

where the last relation can be fulfilled in light of (12.10). So, by Maximum Principle, v  ðxÞ f wðxÞ f const Mð1  jxjÞ Recalling (12.10), we get

in B1  Bk :

174

Valdinoci, Plane-like minimizers in periodic media

v  ðxÞ f const Mð1  jxjÞ in B1 : Rescaling back:   jxj in Bd 0 : vðxÞ f const Md 0 1  d0

ð12:11Þ

We can now follow [AC], p. 112: Fix any y A Bd 0 =2 and consider the following transformation of Bd 0 such that y becomes the new origin: Ty ðxÞ :¼

d 0  jxj y þ x; d0


 and set, for any function f , fy ðxÞ :¼ f Ty ðxÞ . For any z A qB1 , define

rz :¼ inffr A ½d 0 =8; d 0  s:t: UðrzÞ ¼ 0g; if this set is non-empty. For almost all such z we have vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ud u Ð0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ðd 0 vy ðrz zÞ ¼ ‘ðUy  vy ÞðrzÞ
z dr e t j‘ðUy  vy ÞðrzÞj 2 dr
d 0  rz : ð12:12Þ rz

rz

By (12.11), for any r A ½0; d 0 ,

  d 0  r vy ðrzÞ f const M d 0  y þ rz d0   d0  r d0  r ; jyj  r f const M f const M d 0  d0 2 since jyj e d 0 =2. From this and (12.12) M 2 ðd 0  rz Þ e const

ð12:13Þ

Ðd 0

rz

j‘ðUy  vy ÞðrzÞj 2 dr:

Also, using polar coordinates and the definition of rz , Ð

Bd 0 Bd 0 =4 ð yÞ

wfU¼0g e const

e const

Ð

Bd 0 Bd 0 =8

Ð

Ðd 0

qB1 d 0 =8

e d 0n1 const

wfUy ¼0g

r n1 wfUy ¼0g ðrzÞ dr dz ¼ const

Ð

qB1

Ð Ðd 0

qB1 rz

ðd 0  rz Þ dz:

From this, integrating both sides of (12.13), it follows that

r n1 wfUy ¼0g ðrzÞ dr dz

Valdinoci, Plane-like minimizers in periodic media

const M 2

Ð

Bd 0 Bd 0 =4 ð yÞ

wfU¼0g e d 0n1

Ðd 0

Ð

qB1 d 0 =8

Ð

e const

Bd 0

175

j‘ðUy  vy ÞðrzÞj 2 dr dz

j‘ðU  vÞj 2 :

Applying this result to two points y1 and y2 in Bd 0 =2 so that Bd 0 =4 ðy1 Þ and Bd 0 =4 ðy2 Þ do not overlap, and summing up the two estimates, we obtain M2

Ð

Bd 0

wfU¼0g e const

Ð

Bd 0

j‘ðU  vÞj 2 :

Thus, recalling (12.9), and writing U ¼ UG explicitly: M 2 L n ðfUG ¼ 0g X Bd 0 Þ e const L n ðfjuj ¼ 1g X Bd 0 Þ: Since fjuj ¼ 1g O fUþ ¼ 0g W fU ¼ 0g, we deduce that M 2 L n ðfjuj ¼ 1g X Bd 0 Þ e 2 const L n ðfjuj ¼ 1g X Bd 0 Þ: Accordingly, recalling also (12.9), if M were too big, we get juj < 1 in Bd 0 , which is a contradiction. r o Theorem 12.3. If u A MMo , then u A C 0; 1 ðSM Þ. Furthermore, for any compact domain o D well contained in SM with

D X fjuj ¼ 1g 3 j;

ð12:14Þ

o there exists a suitable positive constant CD , depending only on distðD; qSM Þ, n and the bounds on ai; j and F such that

kukC 0; 1 e CD :

ð12:15Þ

Proof. First of all, we show that ‘u is bounded in D X N, where N is a suitable neighborhood of qfjuj ¼ 1g. 1 o distðx; qSM Þ, we are under the hypotheses of 10 Lemma 12.2, so that (setting B :¼ Bdistðx; fjuj¼1gÞ=10 ðxÞ for short) Indeed: if distðx; fjuj ¼ 1gÞ e

ð12:16Þ

sup minf1  uðx0 Þ; 1 þ uðx0 Þg e const distðx; fjuj ¼ 1gÞ:

x0 A B


Also, by Lemma 2.4, qi ai; j qj ð1 G uÞ ¼ 0 in B; therefore, by elliptic estimates, j‘ð1 G uÞðxÞj e const

sup j1 G uj B

distðx; fjuj ¼ 1gÞ

From this and (12.16), we deduce a bound on ‘u in D X N.

:

176

Valdinoci, Plane-like minimizers in periodic media

This and standard elliptic estimates (for instance, [GT], Theorem 8.32 together with a o scaling argument) imply that u A C 0; 1 ðSM Þ. o , with We now prove (12.15): Take D 0 such that D HH D 0 HH SM o distðD 0 ; qSM Þf

9 o Þ: distðD; qSM 10

With no loss of generality, we may assume D and D 0 to be connected. Fix any x A D 0 and 1 o set r0 :¼ distðD; qSM Þ. Consider a covering of D 0 with balls Br0 ðxj Þ, centered in D 0 , with 5 x0 :¼ x and xj A Br0 =2 ðxj1 Þ. Let k be the number of such balls: obviously, k depends only o on n; D 0 and r0 , hence we may think k to depend only on n; D and distðD; SM Þ. By (12.14), there exists k  e k so that Br0 ðxk  Þ X fjuj ¼ 1g 3 j; while Br0 ðxj Þ X fjuj ¼ 1g ¼ j;

Ej < k  :

It follows that distðxk  ; fjuj ¼ 1gÞ e

2 o Þ; distðxk  ; qSM 9

so we can apply Lemma 12.2 and infer that ð12:17Þ

minf1  uðxk  Þ; 1 þ uðxk  Þg e const r0 :


Also, qi ai; j qj ð1 G uÞ ¼ 0 in Br0 ðxj Þ for j < k  because of Lemma 2.4. So, we deduce from Harnack inequality that
 const 1 G uðxjþ1 Þ f 1 G uðxj Þ: Iterating, using (12.17) and recalling that x0 ¼ x, we get ð12:18Þ

sup minf1  uðxÞ; 1 þ uðxÞg e const r0 :

x A D0

Take now z A D, with juðzÞj < 1. Two cases are possible: If distðz; fjuj ¼ 1gÞ f

1 distðD; qD 0 Þ; 3

then 
qi ai; j qj ð1 G uÞ ¼ 0 on BdistðD; qD 0 Þ=3 ðzÞ;

Valdinoci, Plane-like minimizers in periodic media

177

thus, elliptic estimates imply that const distðD; qD 0 Þ

j‘ð1 G uÞðzÞj e

sup BdistðD; qD 0 Þ=3 ðzÞ

j1 G uj:

From this and (12.18), j‘uðzÞj e CD . On the other hand: If distðz; fjuj ¼ 1gÞ
yg X BK Þ e L n ðfu > y ? g X BK Þ; therefore, using the result for y ? , our hypotheses on F and (i), we infer that Cr n e L n ðfu > y ? g X Br Þ e L n ðfu > yg X Br Þ þ L n ðfy ? < u e yg X Br Þ e L n ðfu > yg X Br Þ þ

Ð 1 F inf? F Br

u A ½y ; y0 

e L n ðfu > yg X Br Þ þ Cr n1 ; and so L n ðfu > yg X Br Þ f Cr n . From these consideration, in the rest of the proof, we may and do assume that y is close to 1. We distinguish three cases: Case 1: if 0 < d < 2. We define VðrÞ ¼ L n ðBr X fu > ygÞ; Ð F ðx; uÞ: AðrÞ ¼ Br Xfueyg

We want to prove that ð13:19Þ



n1 C Vðr  1Þ n þ Aðr  1Þ e VðrÞ  Vðr  1Þ þ AðrÞ  Aðr  1Þ:

Indeed, if (13.19) holds, then, by induction5), one sees that VðkÞ þ AðkÞ f Ck n ; recalling (i), we get VðkÞ f Ck n , for k big enough, proving (ii)—notice that we used here the hypothesis VðKÞ f m 0 to start the induction. We now prove (13.19). Let h be an auxiliary function to be chosen later in such a way that 1 e h e 1, h ¼ 1 on qBr and h ¼ 1 in Br1 . We will denote by C suitable constants and by A a free parameter, to be chosen in the sequel. Let s ¼ minfu; hg and b ¼ minfu  s; 1 þ yg. By Sobolev and Cauchy Inequalities, and the fact that ‘b vanishes outside fu  s e 1 þ yg, ð13:20Þ

Ð

Br

b

2n n1

n1 n

Ð

eC

Br Xfuse1þyg

jbj j‘bj

Ð
 e CA ai; j qi uqj u  ai; j qi sqj s  2ai; j qi ðu  sÞqj s Br

þ

Ð C ðu  sÞ 2 : A Br Xfuse1þyg

5) Details of these inductions can be found in [CC]. Here we just mention that, since Vðr  1Þ f m 0 ,
n1
n1 C 1 Vðr  1Þ þ Aðr  1Þ n e Vðr  1Þ n þ Aðr  1Þ:

Valdinoci, Plane-like minimizers in periodic media

179

Integration by parts, together with the minimality property of u, implies ð13:21Þ

Ð

2n

b n1

Br

n1 n

hÐ
 e CA F ðx; sÞ  F ðx; uÞ Br

þ2 þ

i Ð
ai; j qi;2 j sðu  sÞ þ ðqi ai; j Þqj sðu  sÞ

Br

Ð C ðu  sÞ 2 : A Br Xfuse1þyg

Since s ¼ h ¼ 1 in Br1 , the left hand side of (13.21) is bigger than
n1 C L n ðBr1 X fu > ygÞ n :

We now estimate the right hand side of (13.21) in Br1 , in which s ¼ 1: Notice now that, if y is close to 1 and u e y, we have, by our assumptions on F , that F ðx; uÞ f Cð1 þ uÞ d f

ð13:22Þ

C ð1 þ uÞ 2 2 2d

so that Ð Ð C CA ð1 þ uÞ 2  F ðx; uÞ e 0 A Br1 Xfueyg 2 Br1 Xfueyg if we choose A suitably big. Therefore, the right hand side of (13.21) in Br1 is negative and less than CAðr  1Þ. On the integration over Br  Br1 , we split the domain of integration. First of all, since y is close to 1 and s e u, our assumptions on F imply that F ðx; sÞ e F ðx; uÞ if u e y. Thus, CA

Ð

Br Br1


F ðx; sÞ  F ðx; uÞ e CA L n fu > yg X ðBr  Br1 Þ :

Also, recalling (13.22), 
Ð C Ð C C ðu  sÞ 2 e ðu þ 1Þ 2 þ L n fu > yg X ðBr  Br1 Þ A Br Br1 A ðBr Br1 ÞXfueyg A e


Ð C C F ðx; uÞ þ L n fu > yg X ðBr  Br1 Þ : A ðBr Br1 ÞXfueyg A

Finally, we define, d 0 ¼ maxf1; dg and, for r A ½r  1; r, 2

ð13:23Þ

2d 0 hðrÞ ¼ 2ðr  r þ 1Þþ  1:

180

Valdinoci, Plane-like minimizers in periodic media

Notice that this setting is possible for our assumption on d. With this definition, we can complete the estimate on the right hand side of (13.21): Indeed, from a direct calculation one sees that, in Br  Br1 , j‘hj þ jD 2 hj e Cðr  r þ 1Þ

2ðd 0 1Þ 2d 0

e Cðh þ 1Þ d

0 1

:

Also, in the set fu > sg we have s ¼ h and then the last formula implies 0

ðj‘hj þ jD 2 hjÞ
ðu  sÞ e Cðh þ 1Þ d 1
ðu  hÞ

ð13:24Þ

0

e C ðu þ 1Þ d ; in ðBr  Br1 Þ X fu > sg. Therefore, recalling that s ¼ 1 in Br1 , splitting the domain of integration into the sets fu e yg and fu > yg, using (13.24) and our hypotheses on F , we get  Ð
ai; j qi;2 j sðu  sÞ þ ðqi ai; j Þqj sðu  sÞ

Br

¼

Ð

ðBr Br1 ÞXfu>sg


ai; j qi;2 j sðu  sÞ þ ðqi ai; j Þqj sðu  sÞ


e CL n ðBr  Br1 Þ X fu > yg þ C


e CL n ðBr  Br1 Þ X fu > yg þ C

Ð

ðBr Br1 ÞXfueyg

Ð

ðBr Br1 ÞXfueyg

ðu þ 1Þ d

0

F ðx; uÞ:

Collecting all these estimates, the proof of (13.19) follows. This finishes the proof of (ii) in the first case. Case 2: if d ¼ 2. We use suitable positive parameters Y and T: we will fix Y small enough and then choose T suitably large (and in fact YT suitably large). Let k A N. We use a barrier function h ¼ hk A C 2 ðBðkþ1ÞT Þ so that 1 e h e 1, h ¼ 1 on qBðkþ1ÞT , ð13:25Þ

ðh þ 1Þ þ j‘hj þ jD 2 hj e Cðh þ 1Þ e CeYTðkþ1jÞ

in BjT  Bð j1ÞT , and ð13:26Þ

j‘hj þ jD 2 hj e CYðh þ 1Þ

in Bðkþ1ÞT . From the last formula and our assumptions on F , we deduce, if Y is suitably small and h close to 1, ð13:27Þ

pffiffiffiffi j‘hj þ jD 2 hj e YFu ðx; hÞ

in Bðkþ1ÞT . We postpone the explicit construction of such h to the end of the proof. Define y 0 ¼ y  CeYT ; if T is large enough, y 0 > 1. Define also

Valdinoci, Plane-like minimizers in periodic media

181

b ¼ minfu  s; 1 þ y 0 g:

s ¼ minfu; hg and As in (13.20)–(13.21), we have that ð13:28Þ

 Ð

2n

b n1

Bðkþ1ÞT

n1 n

h e CA

Ð

Bðkþ1ÞT Xfu>sg


F ðx; sÞ  F ðx; uÞ þ ai; j qi;2 j sðu  sÞ

þ qi ai; j qj sðu  sÞ

i

þ

Ð C ðu  sÞ 2 ; A Bðkþ1ÞT Xfuse1þy 0 g

where A is a free parameter, to be chosen suitably large in what follows. We now estimate the left hand side of (13.28). From (13.25), if T is big enough, we get y  h f ð1  y0 Þ=2 in n1 BkT . Thus, the left hand side of (13.28) is bigger than CVðkTÞ n . Let us now estimate the right hand side of (13.28). First of all, we consider the contribution in fu e yg. Let us observe that, since 1 e s e u e 1, 1 ðu þ 1Þ 2  ðs þ 1Þ 2  ðu  sÞ 2 2 1 ¼ ðu þ sÞðu  sÞ þ 2ðu  sÞ  ðu  sÞ 2 2   1 3 ¼ ðu  sÞ u þ s þ 2 f 0; 2 2 accordingly, in fs < u e yg, ð13:29Þ

Ðu F ðx; uÞ  F ðx; sÞ ¼ Fu ðx; zÞ dz h

Ðu f C ðz þ 1Þ dz ¼ C½ðu þ 1Þ 2  ðh þ 1Þ 2  h

f Cðu  hÞ 2 :

Consequently, choosing A suitably large and recalling (13.27), the contribution of the right hand side of (13.28) in fu e yg is controlled by ð13:30Þ

Ð

Bðkþ1ÞT Xfs yg. First of all, the contribution in Bðkþ1ÞT  BkT of such term is bounded by ð13:31Þ

Ð

ðBðkþ1ÞT BkT ÞXfu>yg


 F ðx; sÞ  F ðx; uÞ þ ðs þ 1Þðu  sÞ þ ðu  sÞ 2 ;

and this term can be bounded by 
L n fu > yg X ðBðkþ1ÞT  BkT Þ :

Let us now look at the contribution of the right hand side of (13.28) in fu > yg X BkT . Notice that, from (13.25), BkT X fs < u e s þ 1 þ y 0 g O BkT X fs < u e yg: Thus, for the contribution of the right hand side of (13.28) in fu > yg X BkT , we are left with ð13:32Þ

Ð

BkT Xfu>yg

e

F ðx; sÞ  F ðx; uÞ þ j‘hj þ jD 2 hj

k P

Ð

j¼1 BjT Bð j1ÞT Xfu>yg

F ðx; hÞ þ j‘hj þ jD 2 hj:

Notice that, by our assumption on F in the case d ¼ 2, F ðx; 1 þ sÞ e const s; provided s > 0 is small enough. Using this and (13.25), we bound the above term in (13.32) by k P

j¼1


 eYTðkþ1jÞ Vð jTÞ  V ð j  1ÞT :

Collecting all the estimates, we get k




n1 P C VðkTÞ n e V ðk þ 1ÞT  VðkTÞ þ eYTðkþ1jÞ Vð jTÞ  V ð j  1ÞT : j¼1


 Let us now define ak ¼ VðkTÞ  V ðk  1ÞT . Notice that fore C

 P

1e jek

aj

n1 n

e akþ1 þ

k P

P

1e jek

aj ¼ VðkTÞ and there-

1

ee ðkþ1jÞ aj ;

j¼1

with e :¼ 1=ðYTÞ > 0. By induction over k, assuming e small enough, one easily gets that ak f ck n1 and thus VðkTÞ f ck n , which is the desired result for the case d ¼ 2. See [CC], page 10, for a description of the inductive argument in full detail.

Valdinoci, Plane-like minimizers in periodic media

183

We now construct explicitly the barrier introduced at the beginning of Case 2. Define the following functions F : ½0; 1 ! R, C : ½1; ðk þ 1ÞT  ! R: 3 6

FðrÞ ¼ 2e Y½ 8 r

 10 r 4 þ15 r 2 ðkþ1ÞT  8 8

1

and CðrÞ ¼ 2e Y½rðkþ1ÞT   1: By explicit computations, Fð1Þ ¼ Cð1Þ;

F 0 ð1Þ ¼ C 0 ð1Þ and

F 00 ð1Þ ¼ C 00 ð1Þ:

Thus,
the function h agreeing with F in ½0; 1 and with C in ½1; ðk þ 1ÞT  belongs to C 2 ½0; ðk þ 1ÞT ; R . Define hðxÞ ¼ hðjxjÞ. Notice that jF 0 ðrÞj e CYrðF þ 1Þ;

ð13:33Þ

jF 00 ðrÞj e CYðF þ 1Þ

in ½0; 1 and jC 0 ðrÞj þ jC 00 ðrÞj e CYðC þ 1Þ

ð13:34Þ in ½1; ðk þ 1ÞT . Moreover,

  2 jh 0 j þ jh 00 j: ðh þ 1Þ þ j‘hj þ jD 2 hj e ðh þ 1Þ þ 1 þ jxj

ð13:35Þ

By means of (13.33), we bound the right hand side of (13.35) in B1 by CðF þ 1Þ e Ce YðCðkþ1ÞTÞ . Similarly, in BkT  B1 , using (13.34), we bound (13.35) by CðC þ 1Þ e CeYT . This proves (13.25). In a similar way, one can prove (13.26). This finishes the proof of the case d ¼ 2. Case 3: if d ¼ 0. This will be treated similarly to Case 1, but we will be able not to take derivatives of the coe‰cients (so that the results in Part I do not depend on any regularity of ai; j ). Defining s and b as in Case 1, by Sobolev and Cauchy Inequalities, the ellipticity of the coe‰cients and the minimality of u, we deduce that Ð

Br

2n

b n1

n1 n

eC

Ð

Br Xfuse1þyg

 e CA

Ð

Br Xfu>sg

jbj j‘bj


j‘uj 2  j‘sj 2  2‘ðu  sÞ
‘s

Ð C ðu  sÞ 2 A Br Xfuse1þyg  Ð 
e CA Cai; j qi uqj u  j‘sj 2 þ 2ðu  sÞDs þ

Br Xfu>sg

þ

Ð C ðu  sÞ 2 A Br Xfuse1þyg

184

Valdinoci, Plane-like minimizers in periodic media

h e CA

Ð

Br Xfu>sg

þ

i
 Ð j‘sj 2 þ F ðx; sÞ  F ðx; uÞ þ 2 ðu  sÞDs Br

Ð C ðu  sÞ 2 : A Br Xfuse1þyg

The reader may compare this with (13.20) and (13.21). Defining h as in (13.23), we immediately get Ð

Br Xfu>sg


j‘hj 2 e CL n ðBr  Br1 Þ X fu > sg


e CL n ðBr  Br1 Þ X fu > yg þ C

Ð

ðBr Br1 ÞXfueyg

wð1; 1Þ ðuÞ:

From this and the estimates developed for Case 1, the recursive relation (13.19) readily follows. This finishes the proof of the case d ¼ 0. The proof of Theorems 3.1 and 10.1 is completed.

Bibliography [AC] [ACF1] [ACF2] [ACF3] [Ba] [BPV] [BZ] [B] [C] [CC] [CL] [DEF] [G] [GG] [GT] [GL] [GP]

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Universita` di Roma Tor Vergata, Dipartimento di Matematica, Via della Ricerca Scientifica, 00133 Roma, Italy e-mail: [email protected] Eingegangen 15. November 2001, in revidierter Fassung 24. Juli 2003