Communications in PartialDifferential EquationsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713597240

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(1)This article was downloaded by:[EBSCOHost EJS Content Distribution] On: 21 July 2008 Access Details: [subscription number 768320842] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK. Communications in Partial Differential Equations Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713597240. EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R2 FOR A CLASS OF PERIODIC ALLEN-CAHN EQUATIONS Francesca Alessio a; Louis Jeanjean b; Piero Montecchiari c a. Dipartimento di Matematica, Universit. di Torino, Torino, Italy. b. Equipe de Math. de Franche-Comt. c. Dipartimento di Matematica, Universit. matiques, Universit. , Besan on, France. di Ancona, Ancona, Italy. Online Publication Date: 07 January 2002 To cite this Article: Alessio, Francesca, Jeanjean, Louis and Montecchiari, Piero (2002) 'EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R2 FOR A CLASS OF PERIODIC ALLEN-CAHN EQUATIONS', Communications in Partial Differential Equations, 27:7, 1537 — 1574 To link to this article: DOI: 10.1081/PDE-120005848 URL: http://dx.doi.org/10.1081/PDE-120005848. PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material..

(2) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. COMMUN. IN PARTIAL DIFFERENTIAL EQUATIONS, 27(7&8), 1537–1574 (2002). EXISTENCE OF INFINITELY MANY STATIONARY LAYERED SOLUTIONS IN R2 FOR A CLASS OF PERIODIC ALLEN–CAHN EQUATIONS Francesca Alessio,1 Louis Jeanjean,2 and Piero Montecchiari3 1. Dipartimento di Matematica, Universita` di Torino, Via Carlo Alberto, 10, I–10123 Torino, Italy E-mail: alessio@dm.unito.it 2 Equipe de Mathe´matiques, Universite´ de Franche-Comte´ 16, Route de Gray, 25030 Besançon, France E-mail: jeanjean@math.univ-fcomte.fr 3 Dipartimento di Matematica, Universita` di Ancona Via Brecce Bianche, I–60131 Ancona, Italy E-mail: montecch@popcsi.unian.it. ABSTRACT We consider a class of periodic Allen–Cahn equations uðx; yÞ þ aðx; yÞW 0 ðuðx; yÞÞ ¼ 0; 2. ðx; yÞ 2 R2. ð1Þ. 2. where a : R ! R is an even, periodic, positive function and W : R ! R is modeled on the classical two well Ginzburg–Landau potential WðsÞ ¼ ðs2 b2 Þ2 . We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of ð1Þ asymptotic 1537 Copyright & 2002 by Marcel Dekker, Inc.. www.dekker.com.

(3) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1538. ALESSIO, JEANJEAN, AND MONTECCHIARI. as x ! 1 to the pure states b, i.e., solutions satisfying the boundary conditions lim uðx, yÞ ¼ b,. x! 1. uniformly in y 2 R:. ð2Þ. In fact, we prove the existence of solutions of (1)–(2) which are periodic in the y variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of (1)–(2) asymptotic to different periodic solutions as y ! 1. Key Words: Heteroclinic Variational methods. solutions;. Mathematics Subject Classification: 35J20; 34C37. Elliptic. equations;. 35J60; 35B05; 35B40;. 1. INTRODUCTION In this paper we deal with a class of semilinear elliptic equations of the form uðx, yÞ þ aðx, yÞW 0 ðuðx, yÞÞ ¼ 0,. ðx, yÞ 2 R2 ,. ðEÞ. where we assume ðH1 Þ. a is Ho¨lder continuous on R2 , strictly positive and (i) (ii). ðH2 Þ. there exist T1 , T2 > 0 such that aðx þ T1 , y þ T2 Þ ¼ aðx, yÞ for all ðx, yÞ 2 R2 . aðx, yÞ ¼ aðx, yÞ for all ðx, yÞ 2 R2 .. W 2 C2 ðRÞ satisfies (i) (ii). there exists b > 0 such that Wð bÞ ¼ 0, W 00 ð bÞ > 0 and WðsÞ > 0 for any s 2 Rnfb, bg, there exists R0 > b such that WðsÞ > WðR0 Þ for any s > R0 and WðsÞ > WðR0 Þ for any s < R0 :. In the sequel, without loss of generality, we will assume T1 ¼ T2 ¼ 1. This kind of equations arises in various fields of Mathematical Physics and our assumptions on W are modeled on the classical two well Ginzburg–Landau potential WðsÞ ¼ ðs2 1Þ2 . In fact (E) can be viewed as a generalization of the stationary Allen–Cahn equation introduced in 1979.

(4) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1539. by S. M. Allen and J. W. Cahn (see Ref. [6]) as a model for phase transitions in binary metallic alloys. In these models the function u is an order parameter representing pointwise the state of the material. The global minima b of W are called the pure phases and different values of u represent a mixed configuration. In this paper we look for two phases layered solutions of (E). Namely, we look for solutions of (E) asymptotic as x ! 1 to the pure states b, i.e., solutions of the boundary value problem (. uðx, yÞ þ aðx, yÞW 0 ðuðx, yÞÞ ¼ 0, lim uðx, yÞ ¼ b,. x! 1. ðx, yÞ 2 R2 uniformly in y 2 R:. ðPÞ. Apart from its physical aspects, problem (P) presents interesting mathematical features. In Ref. [12], N. Ghoussoub and C. Gui proved a De Giorgi’s conjecture (see Ref. [10]) related to (P). They obtain, in particular, the following result. Theorem 1.1. If aðx, yÞ ¼ a0 > 0 for any ðx, yÞ 2 R2 and if u 2 C2 ðR2 Þ is a solution of (P) then uðx, yÞ ¼ qðxÞ for any ðx, yÞ 2 R2 for some q 2 C2 ðRÞ solution of the problem (. q€ ðxÞ þ a0 W 0 ðqðxÞÞ ¼ 0, lim qðxÞ ¼ b:. x2R. x! 1. In other words, by Theorem 1.1, if a is constant any solution of (P) depends only on the x variable and is solution of the corresponding ordinary differential equation. For other works on this kind of problems as well as extensions of this result in dimension greater than 2 and to more general settings, we refer to Refs. [2,7–9,11] and the references therein. All these results show that if (P) is autonomous then the set of its solutions is in a certain sense trivial, the problem being in fact one dimensional. For completeness we recall that this is not the case, in general, if instead of a single equation one consider systems of autonomous Allen–Cahn equations (Ref. [1]). In the present paper we pursue the work started in Ref. [5] to analyze the set of solutions of (P) in cases where the function a is non constant. In Ref. [5], we proved that, even if a depends only on the x variable (being constant in y), then (P) admits, generically with respect to a, two dimensional solutions, i.e., solutions depending on both the variables x and y. More precisely, we proved the existence of a dense subset (with respect to the L1 -norm) in fa 2 CðRÞ j a is periodic and positiveg for which (P).

(5) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1540. ALESSIO, JEANJEAN, AND MONTECCHIARI. possesses two dimensional solutions. Now we consider a fully non autonomous case and we show: Theorem 1.2. Assume ðH1 Þ–ðH2 Þ. Then, (P) has infinitely many solutions, distinct up to periodic translations. To prove Theorem 1.2, we first consider in Section 2, the periodic problem associated to (P). More precisely, we look for solutions of (P) which are periodic in the y variable, i.e., solutions of 8 0 2 > < uðx, yÞ þ aðx, yÞW ðuðx, yÞÞ ¼ 0, ðx, yÞ 2 R uðx, yÞ ¼ uðx, y þ 1Þ, ðx, yÞ 2 R2 ð1:1Þ > : lim uðx, yÞ ¼ b, uniformly in y 2 R: x! 1. The main feature (and difficulty) of this problem is that it has mixed boundary conditions, requiring the solution to be periodic in the y variable and of the heteroclinic type in the x variable. Letting S0 ¼ R ½0, 1, we look for minima of the action functional Z 1 jruðx, yÞj2 þ aðx, yÞWðuðx, yÞÞ dx dy ’ðuÞ ¼ 2 S0 on the class Z 1. 1 ðS0 Þ juðx, yÞ bj2 dy ! 0 as x ! 1 : ¼ u 2 Hloc 0. As we shall see these minima satisfy the equation and the right boundary conditions in the x variable. In addition they satisfy Neumann boundary conditions in the y variable. To prove that these minima are also solutions of (1.1) we follow P.H. Rabinowitz,[14,15] observing that, thanks to the reversibility assumption ðH1 Þ-ðiiÞ, they belong to the smaller class ( 1 p ¼ u 2 Hloc ðS0 Þ j uðx, 0Þ ¼ uðx, 1Þ for a:e: x 2 R,. Z. 1. ) 2. juðx, yÞ bj dy ! 0 as x ! 1 : 0. To get a minimum of ’ on we use a reduction procedure related to the one we introduced in Ref. [5]. Roughly speaking we consider the elements of as paths x 2 R ! uðx, Þ 2 H 1 ð0, 1Þ and we look for minimal trajectories, for the reduced functional, connecting as x ! 1 the constant.

(6) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1541. functions b. The reduced minimization problem is still characterized by a lack of compactness due to the unboundedness of the domain but now it can be overcome by using concentration compactness type arguments. We want to point out that this same method can be used to obtain directly a solution of (1.1) by minimizing the functional ’ on the class p . Hence, to prove the existence of such minima one does not need the reversibility assumption ðH1 Þ-ðiiÞ. In fact, we use assumption ðH1 Þ-ðiiÞ to prove the existence of solutions to (1.1) which are also minima of ’ on the class . We set c ¼ inf ’. and. . K ¼ fu 2 j ’ðuÞ ¼ cg. and say that two solutions z1 and z2 2 K are equivalent, if there exists j 2 Z such that z1 ðx þ j, yÞ ¼ z2 ðx, yÞ for any ðx, yÞ 2 R2 : We then write z1 x z2 . If the set K= x is infinite then Theorem 1.2 is true. Thus we may assume the opposite. In Section 3, assuming that K= x is finite, or equivalently (see Remark 3.1) that K is constituted by isolated points with respect to the H 1 ðS0 Þ metric, we prove that there exist heteroclinic type solutions of (P) connecting different elements of K as y ! 1. This is done adapting to our setting a method proposed in Refs. [14–16] in the study of the existence of heteroclinic connections between periodic solutions for some class of Lagrangian systems or of semilinear elliptic equations. Precisely, denoting by z , 2 Z, the elements of K, we will look for solutions of (P) which satisfies the boundary conditions lim ku z0 kH 1 ðSj Þ ¼ lim ku z kH 1 ðSj Þ ¼ 0. j!1. j!þ1. where Sj ¼ R ½ j, j þ 1. Such solutions are obtained as local minima of the functional X ’j ðuÞ, ðuÞ ¼ j2Z. where Z ’j ðuÞ ¼. 1 jruðx, yÞj2 þ aðxÞWðuðx, yÞÞ dx dy c: Sj 2. Since, as discussed above, c ¼ min ’, the functional is well defined, and non negative, on

(7). 1 ¼ u 2 Hloc ðR2 Þ j j u 2 , 8 j 2 Z.

(8) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1542. ALESSIO, JEANJEAN, AND MONTECCHIARI. where ð j uÞðx, yÞ ¼ uðx, y þ jÞ. We set . H ¼ u 2 lim ku z0 kH 1 ðSj Þ ¼ lim ku z kH 1 ðSj Þ ¼ 0 j!1 j!þ1 and m ¼ inf ðuÞ and K ¼ fu 2 H j ðuÞ ¼ m g, u2H. 2 Z:. We prove the existence of a 2 Z such that there exists u 2 K . Then u is proved to be in C2 ðR2 Þ and a classical solution of (P). We end the section studying further compactness properties of the minimizing sequences of in H . We say that two solutions u1 and u2 2 K are equivalent, and we write u1 y u2 , if there exists j 2 Z such that u1 ðx, y þ jÞ ¼ u2 ðx, yÞ for ðx, yÞ 2 R2 . In the last section, assuming that K = y is finite, namely that Theorem 1.2 is not already proved, we establish the existence of infinitely many heteroclinic solutions of multibump type. Roughly speaking, for any given N 2 N we show the existence of solutions (infinitely many) of (P) which transit N times between z0 and z while y runs between 1 and þ1. More precisely, for every > 0 and for n 2 N large enough, we set n o U 0 ¼ u 2 j ku z0 kH 1 ðSn Þ and ku z kH 1 ðSn Þ and. n o U ¼ u 2 j ku z kH 1 ðSn Þ and ku z0 kH 1 ðSn Þ :. Then, for every N 2 N, p ¼ ð p1 , . . . , pN Þ 2 ZN and ¼ ð 1 , . . . N Þ 2 f0, gN with pi pi1 4n and i 6¼ i1 for all i ¼ 2, . . . , N, let

(9). HN, p, ¼ u 2 j uðx, y pi Þ 2 U i , for a:e: ðx, yÞ 2 R2 , i ¼ 1, . . . , N and mN, p, ¼ inf HN, p,. and.

(10). KN, p, ¼ u 2 HN, p, jðuÞ ¼ mN, p, :. Thanks to the compactness properties of the minimizing sequences of in H obtained in Section 3, we can adapt to our setting a constrained minimization procedure related to the ones of Refs. [3] and [4] to prove that for any N 2 N, p 2 ZN and 2 f0, gN as above, there exists u 2 KN, p, . Moreover, u 2 C2 ðR2 Þ and it is a solution of (P). This result completes the proof of Theorem 1.2. Using the notation introduced above we can now state more in details our main result..

(11) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1543. Theorem 1.3. Assume ðH1 Þ–ðH2 Þ. Then (P) admits infinitely many solutions distinct up to periodic translations. More precisely we have. (i ) the set K of periodic solutions of (P) is not empty, (ii ) if the set K= x is finite, then there exists 2 Z such that the set K of heteroclinic type solutions of (P) is not empty, (iii ) if the sets K= x and K = y are finite, then for every N 2 N, p ¼ ð p1 , . . . , pN Þ 2 ZN and ¼ ð 1 , . . . , N Þ 2 f0, gN with pi pi1 4n and i 6¼ i1 for all i ¼ 2, . . . , N, the set KN, p, of multibump type solutions of (P) is not empty. Acknowledgments. We wish to thank Professor P. H. Rabinowitz for useful comments and suggestions.. 2. THE PERIODIC PROBLEM In this section, as stated in the introduction, we look for solutions of the problem 8 0 2 > < uðx, yÞ þ aðx, yÞW ðuðx, yÞÞ ¼ 0, ðx, yÞ 2 R uðx, yÞ ¼ uðx, y þ 1Þ, ðx, yÞ 2 R2 ð2:1Þ > : lim uðx, yÞ ¼ b, uniformly in y 2 R: x! 1. Letting S0 ¼ R ½0, 1 we look for minima of the Euler Lagrange functional Z 1 jruðx, yÞj2 þ aðx, yÞWðuðx, yÞÞ dx dy ’ðuÞ ¼ S0 2 on the class n o 1 ¼ u 2 Hloc ðS0 Þ j kuðx, Þ bkL2 ð0, 1Þ ! 0 as x ! 1 : Then we use the reversibility assumption ðH1 Þ-ðiiÞ to show that these minima are indeed minima of ’ on n 1 p ¼ u 2 Hloc ðS0 Þ j uðx, 0Þ ¼ uðx, 1Þ for a:e: x 2 R, o kuðx, Þ bkL2 ð0, 1Þ ! 0 as x ! 1 and so solutions of (2.1). 1 We start this study letting E0 ¼ Hloc ðS0 Þ and noticing that the regularity assumptions on the functions a and W are sufficient to prove that ’ is lower semicontinuous with respect to the weak convergence in E0 , that is,.

(12) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1544. ALESSIO, JEANJEAN, AND MONTECCHIARI. if un ! u weakly in H 1 ðÞ for any relatively compact in S0 , then ’ðuÞ lim inf n!1 ’ðun Þ. Moreover we have Lemma 2.1. If ðun Þ ’ðuÞ then Z. E0 is such that un ! u weakly in E0 and ’ðun Þ ! Z. jrun j2 dx dy !. S0. Z. Z. jruj2 dx dy. and. S0. aðx, yÞWðun Þ dx dy !. aðx, yÞWðuÞ dx dy:. S0. ð2:2Þ. S0. Proof. Indeed Z. Z aðx, yÞWðuÞ dx dy lim inf S0. aðx, yÞWðun Þ dx dy S0. Z lim sup. aðx, yÞWðun Þ dx dy S0. Z. jrun j2 dx dyÞ. ¼ lim supð’ðun Þ S0. Z. jrun j2 dx dy. ¼ ’ðuÞ lim inf S0. Z. aðx, yÞWðuÞ dx dy:. S0. R R Then S0 aðx, yÞWðun Þ dx dy ! S0 aðx, yÞWðuÞ dx dy and, since ’ðun Þ ! ’ðuÞ, (2.2) follows. œ SL ¼ fðx, yÞ 2 S0 j jxj Lg, the two functionals RRemark2 2.1. Letting R S0 jruj dx dy and S0 aðx, yÞWðuÞ dx dy can be written for any L > 0 as sum of two lower semicontinuous functionals as follows Z. 2. Z. 2. jruj dx dy ¼ S0. Z. jruj dy dx þ Z. SL. aðx, yÞWðuÞ dx dy ¼ S0. Z. jruj2 dy dx,. S0 nS L. aðx, yÞWðuÞ dy dx SL. Z aðx, yÞWðuÞ dy dx:. þ S0 nS L.

(13) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1545. Then, arguing as in the preceding lemma, we derive that if ðun Þ that un ! u weakly in E0 and ’ðun Þ ! ’ðuÞ then Z Z jrun j2 dy dx ! jruj2 dy dx, Z Z Z. SL. SL. aWðun Þ dy dx !. S0 nS L. S0 nS L. Z. jrun j2 dy dx !. E0 is such. SL. aWðuÞ dy dx Z Z. aWðun Þ dy dx !. SL. jruj2 dy dx, S0 nS L. aWðuÞ dy dx: S0 nS L. We shall prove that any u 2 E0 with ’ðuÞ < þ1 has some definite asymptotic behavior as x ! 1. To this aim we establish some preliminary results. 1 Note that, by Fubini Theorem, if u 2 E0 then uð, yÞ 2 Hloc ðRÞ for a.e. y 2 ð0, 1Þ. So, if u 2 E0 , for all x1 , x2 2 R we obtain Z 0. 1. 2 Z 1 Z x2 juðx2 , yÞ uðx1 , yÞj dy ¼ @x uðx, yÞ dx dy 0 x1 Z 1 Z x2 2 jx2 x1 j j@x uðx, yÞj dx dy 2. 0. x1. 2’ðuÞjx2 x1 j: Given u 2 E0 , still by Fubini Theorem, the function uðx, Þ 2 L2 ð0, 1Þ for a.e. x 2 R. If ’ðuÞ < þ1, by the previous estimate, the function x ! uðx, Þ is Ho¨lder continuous from a dense subset of R with values in L2 ð0, 1Þ and so it can be extended to a continuous function on R. According to that, any function u 2 E0 \ f’ < þ1g defines a continuous trajectory in L2 ð0, 1Þ verifying 2. Z. 1. dðuðx2 , Þ, uðx1 , ÞÞ ". juðx2 , yÞ uðx1 , yÞj2 dy. 0. 2’ðuÞjx2 x1 j,. 8 x1 , x2 2 R:. ð2:3Þ. Then, we obtain Lemma 2.2. For all r > 0 there exists

(14) r > 0 such that if u 2 E0 satisfies minfkuðx, Þ bkH 1 ð0, 1Þ , kuðx, Þ þ bkH 1 ð0, 1Þ g r for a.e. x 2 ðx1 , x2 Þ.

(15) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1546. ALESSIO, JEANJEAN, AND MONTECCHIARI. then Z. x2 Z 1. 1 jruðx, yÞj2 þ aðx, yÞWðuðx, yÞÞ dy dx x1 0 2 1 1 dðuðx2 , Þ, uðx1 , ÞÞ2 þ

(16) 2r ðx2 x1 Þ 2ðx2 x1 Þ 2

(17) r dðuðx1 , Þ, uðx2 , ÞÞ:. ð2:4Þ. Proof. We define on H 1 ð0, 1Þ the functional Z1 1 jq_ð yÞj2 þ aWðqð yÞÞ dy FðqÞ ¼ 0 2 where a ¼ minR2 aðx, yÞ > 0. Observe that if ðx1 , x2 Þ R and u 2 E0 are such that Fðuðx, ÞÞ

(18) , for a.e. x 2 ðx1 , x2 Þ then Z x2 Z 1 1 2 j@x uðx, yÞj dy þ Fðuðx, ÞÞ dx x1 2 0 Z x2 Z 1 1 j@x uðx, yÞj2 dy dx þ

(19) ðx2 x1 Þ x1 2 0 2 Z 1 Z x2 1 j@x uðx, yÞj dx dy þ

(20) ðx2 x1 Þ 2ðx2 x1 Þ 0 x1 1 dðuðx2 , Þ, uðx1 , ÞÞ2 þ

(21) ðx2 x1 Þ 2ðx2 x1 Þ pffiffiffiffiffiffi ð2:5Þ 2

(22) dðuðx2 , Þ, uðx1 , ÞÞ: By (2.5), to prove the lemma it is sufficient to show that for any r > 0, there exists

(23) r > 0 such that if q 2 H 1 ð0, 1Þ is such that minfkq bkH 1 ð0, 1Þ , kq þ bkH 1 ð0, 1Þ g r then FðqÞ ð1=2Þ

(24) 2r . To this aim, we will prove that for any sequence ðqn Þ H 1 ð0, 1Þ satisfying Fðqn Þ ! 0, there is a subsequence ðqnj Þ, such that qnj ! b or qnj ! b strongly in H 1 ð0, 1Þ. So, let ðqn Þ H 1 ð0, 1Þ be such that Fðqn Þ ! 0. If we prove that minfkqn bkL1 ð0, 1Þ , kqn þ bkL1 ð0, 1Þ g ! 0 as n ! 1 we have done. Indeed, in such a case there exists a subsequence ðqnj Þ of ðqn Þ which Rconverge in 1 L1 ð0, 1Þ to b or b. Assuming e.g. kqnj bkL1 ð0, 1Þ ! 0, since 0 jq_ n j2 dy ! 1 0, we conclude that qnj ! b strongly in H ð0,1Þ: Then, assume by contradiction, that there exists a subsequence of ðqn Þ (still denoted qn ) such that minfkqn bkL1 ð0, 1Þ , kqn þ bkL1 ð0, 1Þ g " > 0. Then, there exists a sequence ð y1n Þ ½0, 1 such that minfjqn ð y1n Þ bj,.

(25) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1547. R1 jqn ð y1n Þ þ bjg ". Also, by ðH2 Þ, since 0 Wðqn Þ k dy ! 0, there exists a sequence ð y2n Þ ½0, 1 such that minfjqn ð y2n Þ bj, jqn ð y2n Þ þ bjg ! 0. Thus, for n 2 N large enough we get Z 2 Z 1 1=2 yn " 2 2 1 1 1=2 2 qn ð yn Þ qn ð yn Þ jq_ ðtÞj dt yn yn jq_ n ðtÞj dt y1n n 2 0 pffiffiffi 2Fðqn Þ1=2 œ. contradicting the assumption Fðqn Þ ! 0. The next lemma is a direct consequence of Lemma 2.2. Lemma 2.3. If u 2 E0 \ f’ < þ1g then there exist 2 f bg such that dðuðx, Þ, Þ ! 0. as. x ! 1:. Proof. First of all note that, by Lemma 2.2, there exist 2 f bg such that lim inf dðuðx, Þ, Þ ¼ 0: x! 1. Considering the case x ! þ1 we argue by contradiction. We assume that there exists r 2 ð0, b=4Þ such that lim supx!þ1 dðuðx, Þ, þ Þ > 2r. Then, by (2.3), there exists a sequence of disjoint intervals ð pi , si Þ, i 2 N, such that dðuð pi , Þ, þ Þ ¼ r, dðuðsi , Þ, þ Þ ¼ 2r and r dðuðx, Þ, þ Þ 2r for x 2 [i ð pi , si Þ. Using (2.4) this implies ’ðuÞ ¼ þ1, a contradiction. Similarly one proves that dðuðx, Þ, Þ ! 0 as x ! 1. œ We define the class ¼ fu 2 E0 \ f’ < þ1g j dðuðx, Þ, bÞ ! 0 as x ! 1g and let c ¼ inf ’ . and. K ¼ fu 2 j ’ðuÞ ¼ cg:. Moreover, for any > 0 we set 1 ¼ 2 þ 2. ! max aðx, yÞ . ðx, yÞ2R2. ! maxpffiffi WðsÞ js bj 2. :. ð2:6Þ. We shall prove that the minimal set K is not empty. In this aim we start noticing that the trajectories in with action close to the infimum satisfy suitable concentration properties..

(26) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1548. ALESSIO, JEANJEAN, AND MONTECCHIARI. Lemma 2.4. There exists 0 2 ð0, b=2Þ such that for any 2 ð0, 0 Þ there exist > 0 and ‘ > 0 for which, if u 2 and ’ðuÞ c þ , then (i) (ii). if minfkuðx, Þ bkH 1 ð0, 1Þ , kuðx, Þ þ bkH 1 ð0, 1Þ g for a.e. x 2 ðs, pÞ, then p s ‘ , if kuðx , Þ þ bkH 1 ð0, 1Þ , then dðuðx, Þ, bÞ for any x x and if kuðxþ ,Þ bkH 1 ð0,1Þ , then dðuðx, Þ, bÞ for any x xþ .. Moreover ! 0 as ! 0. Proof. Note that ðiÞ plainly follows from Lemma 2.2. To prove ðiiÞ, we need þ some additional definition. Since p & ffiffiffiffiffi 0 as ! 0 , for all small enough we can choose r such that

(27) r , where

(28) r is given by Lemma 2.2. þ Moreover, r can be choose such that r & 0 pas ffiffiffiffiffi ! 0 . Let ¼ maxf , r g þ ð3 =

(29) r Þ. Since by definition,

(30) r , we have that ! 0 as ! 0þ . Let 0 2 ð0, b=2Þ be such that < b=2 for all 2 ð0, 0 Þ: Now let 2 ð0, 0 Þ, u 2 and x 2 R be such that kuðx , Þ þ bkH 1 ð0, 1Þ and ’ðuÞ c þ . We define 8 if x < x 1, < b u ðx, yÞ ¼ ðx x Þb þ ðx x þ 1Þuðx , yÞ if x 1 x x : uðx, yÞ ifx x and note that since u 2 , p ’ðu ffiffiffi Þ c. Moreover, by definition (2.6) of , recalling that kqkL1 ð0, 1Þ 2kqkH 1 ð0, 1Þ for any q 2 H 1 ð0, 1Þ, a direct estimate gives Z x Z 1 1 2 jru j þ aWðu Þ dy dx x 1 0 2 from which, since 1 2 jruj þ aWðuÞ dy dx ’ðu Þ ¼ ’ðuÞ 1 0 2 Z x Z 1 1 jru j2 þ aWðu Þ dy dx, þ x 1 0 2 Z. we obtain Z x Z 1. 0. 1. x Z 1. 1 jruj2 þ aWðuÞ dy dx 2 : 2. ð2:7Þ. Now, assume by contradiction that there exists x1 < x such that dðuðx1 , Þ, bÞ . Then there exists x2 2 ðx1 , x such that.

(31) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1549. dðuðx, Þ, bÞ maxf , r g for x 2 ðx1 , x2 Þ and dðuðx1 , Þ, uðx2 , ÞÞ maxf , r g. By (2.4), we get Z. x Z 1 1. 0. 1 jruj2 þ aWðuÞ dy dx

(32) r ð maxf , rd gÞ 3 2. which contradicts (2.7). This proves that dðuðx, Þ, bÞ < for any x x . Analogously, if kuðxþ , Þ bkH 1 ð0, 1Þ then dðuðx, Þ, bÞ < for any x xþ . œ According to Lemma 2.4, we fix 2 ð0, 0 Þ such that ¼ b=6 and we denote ¼ and ‘ ¼ ‘ . To exploit compactness properties of ’ in , it is useful to introduce the function X : E0 ! R [ fþ1g given by. b XðuÞ ¼ sup x 2 R j dðuðx, Þ, bÞ : 2 We can now describe a first compactness property of the minimizing sequences of ’ in . Lemma 2.5. If ðun Þ is such that ’ðun Þ ! c and Xðun Þ ! X0 2 R, then there exists u0 2 K such that, along a subsequence, kun u0 kH 1 ðS0 Þ ! 0. Proof. To prove the lemma we need some preliminary estimates. (1). For all > 0 there exists R > 0 such that if u 2 \ f’ c þ g then. kuðx, ÞkL2 ð0, 1Þ R. 8 x 2 R:. ð2:8Þ. Indeed, let Br ¼ fq 2 L2 ð0, 1Þ j kqkL2 ð0, 1Þ rg and note that B2b contains as interior points the functions b. To prove (2.8), assume by contradiction that there exists > 0 such that for any R > 2b there exists u 2 \ f’ c þ g such that uðx , Þ 62 BR for some x 2 R. Since dðuðx, Þ, bÞ ! 0 as x ! 1, by continuity (see (2.3)), there exists x such that uðx , Þ 2 @B2b , uðxþ , Þ 2 @BR and 2b < kuðx, ÞkL2 ð0, 1Þ < R for all x 2 ðx , xþ Þ. Then minfdðuðx, Þ, bÞ, dðuðx, Þ, bÞg b for all x 2 ðx , xþ Þ and by Lemma 2.2 we obtain c þ ’ðuÞ

(33) b dðuðx , Þ, uðxþ , ÞÞ

(34) b ðR 2bÞ which is a contradiction for R large enough..

(35) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1550. ALESSIO, JEANJEAN, AND MONTECCHIARI. For any " > 0 there exist ð"Þ 2 ð0, Þ and ‘ð"Þ > ‘ such that if u 2 \ f’ c þ ð"Þg then Z 1 2 jruj dy dx " and. (2) Z. 0. jxXðuÞj‘ð"Þ. Z. Z. jxXðuÞj‘ð"Þ. 1. WðuðxÞÞ dy dx ":. ð2:9Þ. 0. Indeed, by Lemma 2.4, let < be such that ð"=3Þ minf1=2, a g, where a ¼ maxR2 a. Then, given any u 2 \ f’ c þ g, by Lemma 2.4, there exist x 2 ðXðuÞ 2‘ , XðuÞÞ and xþ 2 ðXðuÞ, XðuÞ þ 2‘ Þ such that kuðx , Þ þ bkH 1 ð0, 1Þ and kuðxþ , Þ bkH 1 ð0, 1Þ . We define the function 8 b if x < x 1, > > > > < ðx x Þb þ ðx x þ 1Þuðx , yÞ if x 1 x x if x x xþ u ðx, yÞ ¼ uðx, yÞ > > > ðxþ þ 1 xÞuðxþ , yÞ þ ðx xþ Þb if xþ x xþ þ 1 > : b if x > xþ þ 1 and since ’ðu Þ c, as in the proof of Lemma 2.4, one obtains that Z 1 Z 1 jruj2 þ aWðuÞ dy dx ’ðuÞ ’ðu Þ þ 2 3 : 0 2 jxXðuÞj2‘ þ1 Then (2.9) follows setting ‘ð"Þ ¼ 2 maxf‘ , ‘ g þ 1 and ð"Þ ¼ . (3) There exists > 0 such that for all " > 0, u 2 \ f’ c þ ð"Þg we have Z 1 Z 2 juðx, yÞ Uðx XðuÞ, yÞj dy dx " jxXðuÞj‘ð"Þ. for. all. ð2:10Þ. 0. where 8 < b Uðx, yÞ ¼ ðx 1Þb þ xb : b. if x < 0, if 0 x 1 if x > 1:. First, by ðH2 Þ, note that both b are non degenerate minima of W and around each of these points W behaves quadratically. Thus, setting dðsÞ ¼ minfjs bj, js þ bjg, we have b there exists w0 > 0 such that WðsÞ w0 d 2 ðsÞ when dðsÞ : 2. ð2:11Þ.

(36) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1551. Now, given u 2 \ f’ c þ ð"Þg, let S ¼ ð1, XðuÞ ‘ð"Þ ½0, 1 and A ¼ fðx, yÞ 2 S j juðx, yÞ þ bj > b=2g. Then, since dðuðx, yÞÞ ¼ juðx, yÞ þ bj for all ðx, yÞ 2 S nA , by (2.11) and (2.9) it follows Z Z 1 " 2 juðx, yÞ þ bj dx dy Wðuðx, yÞÞ dx dy : ð2:12Þ w w 0 S nA 0 S nA R 1 To estimate S nA juðx, yÞ þ bj2 dx dy, we note that since u 2 Hloc ðS0 Þ there exists C ð0, 1Þ with measðCÞ ¼ 1 and such that the function x ! uðx, yÞ is continuous on R for any y 2 C. Then, setting Ay ¼ fx < XðuÞ ‘ð"Þ j juðx, yÞ þ bj > b=2g, we have that Ay is open for any y 2 C and so A ¼ [y2CRAy is open too. Hence, measðA nðA R R since 1 2 ð0, 1ÞÞÞ ¼ 0, we have A juðx, yÞ þ bj dx dy A 0 juðx, yÞ þ bj2 dy dx. We then observe that, for any x 2 A , since x XðuÞ ‘ð"Þ XðuÞ ‘ , by Lemma 2.4 we have dðuðx, Þ, bÞ b=4 and so there exists yx 2 ð0, 1Þ such that juðx, yx Þ þ bj b=4. Moreover, for all x 2 A , by definition, there exists yx such that ðx, yx Þ 2 A . Since, by Fubini’s Theorem uðx, Þ 2 H 1 ð0, 1Þ for a.e. x 2 A , we obtain that for a.e. x 2 A Z x y b x juðx, y Þ uðx, yx Þj j@y uðx, yÞj dy yx 4 Z 1 1=2 jyx yx j1=2 j@y uðx, yÞj2 dy 0. and thus Z1. j@y uðx, yÞj2 dy . 0. b2 : 16. Now since, by (2.9), Z Z 1 Z j@y uj2 dy dx A. 0. jruj2 dx dy ". S. we deduce that measðA Þ ð16=b2 Þ". Then, it follows Z Z Z 1 2 2 juðx, yÞ þ bj dx dy juðx, yÞ þ bj dy dx A. A. 0. measðA Þdðuðx, Þ, bÞ2 ": Taking into account (2.12) we obtain Z juðx, yÞ þ bj2 dx dy ", 2 S. ð2:13Þ.

(37) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1552. ALESSIO, JEANJEAN, AND MONTECCHIARI. where ¼ 2ðð1=w0 Þ þ 1Þ. In the same way we can prove that Z juðx, yÞ bj2 dx dy " 2 Sþ. ð2:14Þ. where Sþ ¼ ½XðuÞ þ ‘ð"Þ, þ1Þ ½0, 1. Gathering (2.13) and (2.14), (2.10) follows. We can now prove the lemma. Let ðun Þ be such that ’ðun Þ ! c and Xðun Þ ! X0 2 R. Using (2.8) we have that there exists R > 0 such that Rkun ðx, Þk2L2 ð0, 1Þ R for all x 2 R, n 2 N. Since we also 1 have S0 jrun j dx dy 2’ðun Þ we conclude that ðun Þ is bounded in Hloc ðS0 Þ. 1 Thus, there exists u0 2 Hloc ðS0 Þ such that, up to a subsequence, un ! u0 1 weakly in Hloc ðS0 Þ: We shall now prove that u0 2 and kun u0 kH 1 ðS0 Þ ! 0. By (2.9) and (2.10) we derive that for any " > 0 there exists ‘ð"Þ > ‘. for which Z 1 Z 2 jun ðx, yÞ Uðx X0 , yÞj dy dx " ð2:15Þ jxX0 j‘ð"Þ. 0. for any n 2 N sufficiently large. Then, by semicontinuity, we deduce Z 1 Z ju0 ðx, yÞ Uðx X0 , yÞj2 dy dx ": ð2:16Þ jxX0 j‘ð"Þ. 0. The inequality (2.16) implies, by Lemma 2.3, that u0 2 and then ’ðu0 Þ c. By semicontinuity again we obtain ’ðu0 Þ ¼ c, i.e., u0 2 K. To prove kun u0 kH 1 ðS0 Þ ! 0 we note that since un u0 ! 0 strongly in L2loc ðS0 Þ, the arbitrariness of " in (2.15) and (2.16) implies that un u0 ! 0 strongly in L2 ðSR0 Þ. Moreover byR Lemma 2.1, since ’ðun Þ ! c ¼ ’ðu0 Þ, we obtain S0 jrun j2 dx dy ! S0 jru0 j2 dx dy which, together with the fact that run ! ru0 weakly in L2 ðS0 Þ, implies kun u0 kH 1 ðS0 Þ ! 0. œ As direct consequence of Lemma 2.5, we obtain the following estimate which will be useful in the next section. Lemma 2.6. For all r > 0 there exists hr > 0 such that if u 2 and ’ðuÞ c þ hr then inf z2K ku zkH 1 ðS0 Þ r. Proof. Arguing by contradiction, let r > 0 and ðun Þ be such that ’ðun Þ ! c and inf z2K kun zkH 1 ðS0 Þ > r for all n 2 N. Then, setting vn ðx, yÞ ¼ un ðx þ ½Xðun Þ, yÞ (where ½ denote the integer part), by periodicity.

(38) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1553. we have ðvn Þ , ’ðvn Þ ! c and inf z2K kvn zkH 1 ðS0 Þ > r for all n 2 N. This is in contradiction with Lemma 2.5 since ðXðvn ÞÞ ½0, 1. œ Moreover, we can prove the following existence result. Proposition 2.1. K 6¼ ; and any u 2 K satisfies u 2 C2 ðR2 Þ, is a classical solution of u þ aW 0 ðuÞ ¼ 0 on S0 with @y uðx, 0Þ ¼ @y uðx, 1Þ ¼ 0 for all x 2 R and kukL1 ðS0 Þ R0 . Finally, uðx, yÞ ! b as x ! 1, uniformly in y 2 ½0, 1: Proof. We note that by periodicity, there always exists a minimizing sequence ðun Þ for ’ such that Xðun Þ 2 ½0, 1 for any n 2 N. Then, by Lemma 2.5, the set K is not empty. To complete the proof we show that any u 2 K is a classical solution of u þ aW 0 ðuÞ ¼ 0 on S0 satisfying the above specified properties. First, note that if u 2 K, then, by ðH2 Þ, kukL1 ðS0 Þ R0 . Indeed, suppose that u 2 satisfies juj > R0 on a set of non zero measure. Then, setting u~ ¼ maxfminfu, R0 g, R0 g we have Wðu~ Þ < WðuÞ on this set and, using Stampacchia inequality, we deduce that ’ðu~ Þ < ’ðuÞ. Clearly u~ 2 and so this contradict the definition of c. Let h 2 C01 ðS0 Þ and 2 R. Then, u þ h 2 and since u 2 K, ’ðu þ hÞ is a C 1 function of with a local minimum at ¼ 0. Therefore ’0 ðuÞh ¼. Z. rurh þ aW 0 ðuÞh dx dy ¼ 0 S0. for all such h, namely u is a weak solution of u þ aW 0 ðuÞ ¼ 0 on R ð0, 1Þ. Standard regularity arguments, then show u is a classical solution and satisfies the Neumann boundary conditions. Moreover, since kukL1 ðS0 Þ R0 , there exists C > 0 such that kukC2 ðS0 Þ C. This C 2 estimate implies that u satisfies the remaining boundary conditions. Indeed, assume by contradiction that u does not verify uðx, yÞ ! b as x ! 1 uniformly with respect to y 2 ½0, 1. Then, there exist > 0 and a sequence ðxn , yn Þ 2 S0 with xn ! 1 and juðxn , yn Þ þ bj 2 for all n 2 N. The C 2 estimate above implies that there exists > 0 such that juðx, yÞ þ bj . 8 ðx, yÞ 2 B ðxn , yn Þ, n 2 N:. ð2:17Þ. Along a subsequence, yn ! y0 and thus juðx, yÞ þ bj for all ðx, yÞ 2 B=2 ðxn , y0 Þ, n large enough. But this is not possible since u 2 implies that dðuðx, Þ, bÞ ! 0 as x ! 1. One argues analogously to prove the other case. œ.

(39) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1554. ALESSIO, JEANJEAN, AND MONTECCHIARI. We shall now explore the consequence of the reversibility condition ðH1 Þ-ðiiÞ, and in particular we will prove that the solution that we find in Proposition 2.1 is in fact a solution of (2.1). Let cp ¼ inf p ’ where p ¼ fu 2 \ f’ < þ1g j uðx, 0Þ ¼ uðx, 1Þ for a:e: x 2 Rg: Then we have Lemma 2.7. cp ¼ c. Proof. Since p , cp c. To prove the equality suppose cp > c. Then, there exists u 2 such that ’ðuÞ < cp . Writing Z Z 1=2 Z Z 1 1 1 jruj2 þ aWðuÞ dy dx þ jruj2 þ aWðuÞ dy dx ’ðuÞ ¼ 2 R R 0 1=2 2 ¼ I1 þ I2 it follows that minfI1 , I2 g < cp =2. Suppose e.g., I1 < cp =2. Define 8 1 > < uðx, yÞ if x 2 R and 0 y , 2 vðx, yÞ ¼ > : uðx, 1 yÞ if x 2 R and 1 y 1: 2. ð2:18Þ. Then v 2 p and, because of ðH1 Þ-ðiiÞ, ’ðvÞ ¼ 2I1 < cp , contradicting the definition of cp . œ We shall now prove, following Ref. [15], that any u 2 K is periodic in y. Lemma 2.8. If u 2 K then uðx, 0Þ ¼ uðx, 1Þ for all x 2 R. Proof. Suppose u 2 K and define v as in (2.18). Then, v 2 p and it satisfies vðx, yÞ ¼ vðx, 1 yÞ for all ðx, yÞ 2 S0 . Moreover, by ðH1 Þ-ðiiÞ, ’ðuÞ ¼ ’ðvÞ ¼ c ¼ cp and so v 2 K. Then, both u and v are local minimum of ’ and hence both are classical solution of. on S0 u þ aW 0 ðuÞ ¼ 0, ð2:19Þ @y uðx, 0Þ ¼ @y uðx, 1Þ ¼ 0, for all x 2 R: Then, letting w ¼ u v, we obtain. wðx, yÞ ¼ bðx, yÞwðx, yÞ, on S0 @y wðx, 0Þ ¼ @y wðx, 1Þ ¼ 0, for all x 2 R.

(40) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1555. where 8 < aðx, yÞ½Wðvðx, yÞÞ Wðuðx, yÞÞ bðx, yÞ ¼ uðx, yÞ vðx, yÞ : aðx, yÞW 0 ðuðx, yÞÞ. if uðx, yÞ 6¼ vðx, yÞ, if uðx, yÞ ¼ vðx, yÞ:. Note that b is continuous and wðx, yÞ ¼ 0 for ðx, yÞ 2 R ½0, 1=2. But then a local unique continuation theorem for elliptic equations (see e.g., Nirenberg[13]) and a continuation argument imply wðx, yÞ ¼ 0 for ðx, yÞ 2 R ½0, 1, that is u ¼ v in S0 . œ From the above considerations it is immediate to derive the existence of solutions of (2.1). Indeed we know, by Proposition 2.1, that the set K is non empty and that given u 2 K, by Proposition 2.1 and Lemma 2.8, we have uðx, 0Þ ¼ uðx, 1Þ and @y uðx, 0Þ ¼ @y uðx, 1Þ ¼ 0 for all x 2 R. Hence we can extend u by periodicity on R2 , obtaining that u is a classical solution of (2.1). In the sequel, the extension of K by periodicity on R2 will be still denoted by K. We end this section fixing some notation. First note that by the definition of and the proof of Lemma 2.5 the following metric is well defined on Dðu, vÞ0 ¼ ku vkH 1 ðS0 Þ ,. 8 u, v 2 :. Moreover, given u 2 and A , we set Distðu, AÞ0 ¼ inffDðu, vÞ0 j v 2 Ag. 1 Analogously, setting Sj ¼ R ½ j, j þ 1, j 2 Z, for u, v 2 Hloc ðR2 Þ and for 1 2 A Hloc ðR Þ, we set Dðu, vÞj ¼ ku vkH 1 ðSj Þ. and. Dðu, vÞj, k ¼ ku vkH 1 ðSj [:::[Sk Þ ,. and Distðu, AÞj ¼ inf Dðu, vÞj v2A. and. Distðu, AÞj, k ¼ inf Dðu, vÞj, k , v2A. 1 when the quantities make sense. Finally, for u 2 Hloc ðR2 Þ and j 2 Z we denote ð j uÞðx, yÞ ¼ uðx, y þ jÞ.. 3. HETEROCLINIC TYPE SOLUTIONS In the previous section we proved the existence of solutions of (P) which are periodic in the y variable. We say that two solutions z1 and z2 2 K are equivalent, and we write z1 x z2 , when there exists j 2 Z such that z1 ðx þ j, yÞ ¼ z2 ðx, yÞ for any ðx, yÞ 2 R2 . If the set K= x is infinite then.

(41) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1556. ALESSIO, JEANJEAN, AND MONTECCHIARI. Theorem 1.2 holds true. Thus in the rest of the paper we analyze the case where ðÞ. K= x is finite.. In this section, adapting a variational approach introduced in Refs. [14] and [15], we prove that when ðÞ holds (P) admits heteroclinic type solutions connecting different periodic solutions as y ! 1. By ðÞ, K is a countable set of isolated points that we will enumerate by z , 2 Z. Also, setting ¼ inf Dðz , z Þ0 6¼. we have > 0 since, by ðÞ, K is locally finite. Remark 3.1. Assuming ðÞ is equivalent to ask K to be constituted by isolated points with respect to the H 1 ðS0 Þ metric. Indeed if K= x is not finite, there exists a sequence ðun Þ K such that un 6¼ um if n 6¼ m and Xðun Þ 2 ½0, 1Þ for any n 2 N. By Lemma 2.5 there exists u0 2 K such that along a subsequence un ! u0 with respect to the H 1 ðS0 Þ distance and so u0 is not isolated in K. Let

(42). 1 ¼ u 2 Hloc ðR2 Þ j j u 2 , 8 j 2 Z and for u 2 , j 2 Z, define Z 1 jruðx, yÞj2 þ aðxÞWðuðx, yÞÞ dx dy c: ’j ðuÞ ¼ 2 Sj We observe that, ’j ðuÞ ¼ ’0 ð j uÞ ¼ ’ð j uÞ c. Also, by definition of , ’j ðuÞ 0 for all u 2 , j 2 Z and ’j ðuÞ ¼ 0 if and only if j u 2 K: We consider on the functional X ’j ðuÞ ðuÞ ¼ j2Z. and following Ref. [15], we will look for non periodic solutions of (E) as minima of on suitable subsets of . We first note that since each ’j is non negative and lower semicontin1 uous along weakly convergent sequences in Hloc ðR2 Þ, the same holds true for the functional . 1 Lemma 3.1. Let ðun Þ and u 2 be such that un ! u weakly in Hloc ðR2 Þ, then ðuÞ lim inf n!1 ðun Þ..

(43) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1557. Proof. For any fixed k 2 N we have lim inf n!1. k X. ’j ðun Þ . j¼k. k X. ’j ðuÞ:. j¼k. Then, if ðuÞ ¼ þ1 we have also lim inf n!1 ðun Þ ¼ þ1.PIf otherwise ðuÞ < þ1 then, given any " > 0 and fixed k 2 N such that j jj>k ’j ðuÞ < ", we obtain lim inf ðun Þ lim inf n!1. n!1. k X j¼k. ’j ðun Þ . k X. ’j ðuÞ ðuÞ ". j¼k. and the semicontinuity follows since " is arbitrary.. œ. Concerning the coerciveness of , we first establish some estimates which will be useful to characterize the compactness properties of sublevels of . Lemma 3.2. For any r > 0 there exists r > 0 such that if u 2 and ’j ðuÞ þ ’jþ1 ðuÞ r , j 2 Z, then Distðu, KÞj, jþ1 r. Proof. The properties of the functional ’j þ ’jþ1 are very similar to the ones of the functional ’ and to prove that ’j þ ’jþ1 attains its minimum on we can repeat the reasoning of the previous section. Moreover, as in Lemmas 2.7, 2.8, one can show that for any j 2 Z the infimum of the functional ’j þ ’jþ1 on is reached on K. More precisely, ’j ðuÞ þ ’jþ1 ðuÞ ¼ inf v2 ’j ðvÞ þ ’jþ1 ðvÞ if and only if u 2 K. Then, the proof of Lemma 2.6 can be adapted to yield the result. œ According to the previous lemma, we fix > 0 such that if u 2 and Distðu, KÞj, jþ1. j 2 Z verify ’j ðuÞ þ ’jþ1 ðuÞ then : 4. We say that Z is a set of consecutive elements of Z if and only if Z is of the form fh 2 Z j j h < j þ kg or fh 2 Z j j k < h jg for a j 2 Z and k 2 N [ fþ1g. Note that such sets may contain an infinite number of elements. Lemma 3.3. For any u 2 and any set Z of consecutive elements of Z which satisfy ’j ðuÞ =2 for any j 2 Z, there exists 2 Z such that Dðu, z Þj =4, for all j 2 Z..

(44) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1558. ALESSIO, JEANJEAN, AND MONTECCHIARI. Proof. Let j 2 Z be such that j þ 1 2 Z. Then ’j ðuÞ þ ’jþ1 ðuÞ , and thus, by definition of , Distðu, KÞj, jþ1 =4: In particular, by definition of , there exists j 2 Z such that Dðu, zj Þj, jþ1 =4 and therefore Dðu, zj Þj =4 and Dðu, zj Þjþ1 =4. Clearly, if j þ 2 2 Z, then j ¼ jþ1 and the lemma follows. œ Lemma 3.3 allows us to get a first boundedness property of the functions in the sublevels of . Lemma 3.4. For any > 0 there exists

(45) > 0 such that if u 2 f g then k j u k ukL2 ðS0 Þ

(46) for any j, k 2 Z. Proof. First note that if ðuÞ then k j u ð j þ 1Þ ukL2 ðS0 Þ ð2ð þ 2cÞÞ1=2 for any j 2 Z. Indeed we have Z k j u ð j þ 1Þ uk2L2 ðS0 Þ ¼ juðx, y þ 1Þ uðx, yÞj2 d_y dx Sj. and so there exists y in ð j, j þ 1Þ such that Z juðx, y þ 1Þ uðx, y Þj2 dx k j u ð j þ 1Þ uk2L2 ðS0 Þ : R. On the other hand jþ2 Z. Z 2ððuÞ þ 2cÞ j. Z Z j@y uðx, yÞj2 dx dy . R. Z Z. R. 2. y þ1. j@y uðx, yÞj dy. R. y. Z. y þ1. j@y uðx, yÞj2 dy dx. y. juðx, y þ 1Þ uðx, y Þj2 dx. dx R. and thus, by the choice of y , we get k j u ð j þ 1Þ uk2L2 ðS0 Þ 2ððuÞ þ 2cÞ: Now, let u 2 be such that ðuÞ . We denote by LðuÞ the subset of Z for which ’j ðuÞ =2 if j 2 LðuÞ: Plainly, the number lðuÞ of elements of LðuÞ is at most ½2= þ 1 where ½ denote the integer part. Then, the set ZnLðuÞ is constituted of l ðuÞ sets of consecutive elements of Z, Z i ðuÞ, with l ðuÞ lðuÞ þ 1. We have, for any k, j 2 Z, kk u j ukL2 ðS0 Þ ðlðuÞ þ 1Þ sup k j u ð j þ 1Þ ukL2 ðS0 Þ j2Z. þ. l ðuÞ X. sup k j u k ukL2 ðS0 Þ. i¼1 j, k2Z i ðuÞ. ð3:1Þ.

(47) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1559. Then, the first term in the right side of (3.1) is bounded by ðlðuÞ þ 1Þ ð2ð þ 2cÞÞ1=2 and by Lemma 3.3 the sum by l ðuÞ=2. Therefore the lemma follows. œ The following lemma describes a first compactness property of the functional .. < þ1 and Lemma 3.5. Let ðun Þ \ f g be such that there exist

(48). , for a jn 2 Z. Then, there exists 2 Z for which, for all n 2 N, Dðun , z Þjn

(49) 1 u 2 \ f g such that, up to a subsequence, un ! u weakly in Hloc ðR2 Þ. Proof. By Lemma 3.4, there exists

(50) > 0 such that if u 2 \ f g then. , for some kk u k0 ukL2 ðS0 Þ

(51) for any k0 , k 2 Z and if Dðu, z Þj

(52). þ

(53) for any k 2 Z. j 2 Z, we obtain ku z kL2 ðSk Þ

(54) Thus, given ðun Þ \ f g as in the statement, for any L 2 N, setting TL ¼ R ðL, LÞ we obtain. Þ2 þ 2Lc þ : kun z k2L2 ðTL Þ þ krun k2L2 ðTL Þ 2Lð

(55) þ

(56) and we deduce that ðun z Þ is bounded in H 1 ðTL Þ for any L 2 N. This implies that there exists a subsequence ðunk Þ of ðun Þ and a function u such 1 that u z 2 \L>0 H 1 ðTL Þ and unk z ! u z weakly in Hloc ðR2 Þ. Since 1 1 u z 2 \L>0 H ðTL Þ we have that u z 2 H ðSj Þ for any j 2 Z from which we deduce that u 2 and ðuÞ . œ From Lemma 3.5 we also deduce the following result concerning the asymptotic behavior of the functions in the sublevels of . Lemma 3.6. If u 2 \ f < þ1g, there exist 2 Z such that Dðu, z Þj ! 0 as. j ! 1:. Proof. Since ðuÞ < þ1, ’j ðuÞ ! 0 as j jj ! 1 and there exists j0 such that ’j ðuÞ =2 for any j jj j0 . Thus, by Lemma 3.3, there exists 2 Z such that Dðu, zþ Þj =4 for j j0 and Dðu, z Þj =4 for j j0 . Using again that ’j ðuÞ ! 0 as j jj ! 1, we can conclude because of Lemma 2.6. œ By Lemma 3.6 we can restrict ourselves to consider the elements in which have prescribed limits as j ! 1. For 2 Z, we consider the class. H ¼ u 2 lim Dðu, z0 Þj ¼ lim Dðu, z Þj ¼ 0 : j!1. j!þ1.

(57) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1560. ALESSIO, JEANJEAN, AND MONTECCHIARI. By Lemma 3.5 every sequence in H \ f g admits a subsequence which converge in the specified sense to some u 2 and along which the functional is lower semicontinuous. The next step in our proof is to show that there exist particular classes H on which suitable sublevels of satisfy additional compactness properties. We define m ¼ inf ðuÞ and K ¼ fu 2 H j ðuÞ ¼ m g, u2H. 2 Z:. Using suitable test functions one can prove that m < þ1 for any 2 Z. Moreover Lemma 3.7. There results m =2 for any 6¼ 0 and m ! þ1 as jj ! 1. Proof. We observe that Dðz0 , z Þ0 ! þ1 as jj ! 1 and thus, from the definition of H and Lemma 3.4, it readily follows that m ! þ1 as jj ! 1. To prove the first estimate let 6¼ 0 and u 2 H . We have Dðu, z0 Þj ! 0 as j ! 1 while lim inf j!þ1 Dðu, z0 Þj Dðz0 , z Þ0 : Thus, by Lemma 3.3, there exists k0 2 Z such that ’k0 ðuÞ =2 and hence ðuÞ ’k0 ðuÞ =2: œ By Lemma 3.7 there exists 2 Z such that m ¼ min m : 6¼0. As we will see in the next lemma, the minimality property of allows us to further characterize the functions in H whose action is close to m . Remark 3.2. For u 2 , 2 Z and j 2 Z we define 8 < uðx, yÞ uðx, yÞð j þ 1 yÞ þ z ðx, yÞð y jÞ þ ðuÞðx, yÞ ¼ , j : z ðx, yÞ and. if y 2 ð1, j, if y 2 ð j, j þ 1Þ, if y 2 ½ j þ 1, þ1Þ. 8 if y 2 ð1, j, < z ðx, yÞ uðx, yÞð y jÞ þ z ðuÞðx, yÞ ¼ ðx, yÞð j þ 1 yÞ if y 2 ð j, j þ 1Þ, , j : uðx, yÞ if y 2 ½ j þ 1, þ1Þ:. Note that, by continuity, for any > 0 there exists ~ , with ~ ! 0 as ! 0, such that, for any u 2 and j 2 Z, if Dðu, z0 Þj , then ’j ðuÞ ~ . Then, by.

(58) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1561. definition, one can easily see that the following estimate holds. If u 2 is such that Dðu, z Þj for some 2 Z and j 2 Z, then s X. ’k ð , j ðuÞÞ . k¼p. s X. ’k ðuÞ þ 2~. k¼p. for all p < j < s 1. Then we obtain Lemma 3.8. There exists ~0 2 ð0, =4Þ such that for any 2 ð0, ~0 Þ there exist ~ > 0 and ‘~ 2 N for which, if u 2 H and ðuÞ m þ ~ , then (i ) if Distðu, KÞj for all j in a set Z of consecutive numbers, then CardðZÞ ‘~ , (ii ) if Dðu, z0 Þj0 , then Dðu, z0 Þj ~ , 8 j j0 , (iii ) if Dðu, z Þj0 , then Dðu, z Þj ~ , 8 j j0 , (iv ) if 2 Znf0, g, then Dðu, z Þj > , 8 j 2 Z. Moreover ~ # 0 as ! 0þ . Proof. Note that ðiÞ plainly follows from Lemma 2.6. To prove ðiiÞ–ðivÞ we shall need some additional definitions. We fix ~0 2 ð0, =4Þ such that ~ < m =4 for all 2 ð0, 2 ~0 Þ. Moreover, by Lemma 2.6, it is also possible to choose a function f such that, for any u 2 and j 2 Z if ’j ðuÞ then Distðu, KÞj f ð Þ with f ð Þ ! 0 as ! 0. Clearly it is not restrictive to assume that f is a non decreasing function. Then, for any 2 ð0, ~0 Þ we set ~ ¼ f ð2~ Þ. Let u 2 H be such that ðuÞ m þ ~ and assume that j0 2 Z is ~ 2 H . Then, such that Dðu, z0 Þj0 . We define u~ ¼ 0, j0 ðuÞ, noting that u we obtain. m ðu~ Þ ¼ ðuÞ . j0 X. ’k ðuÞ þ ’j0 ðu~ Þ:. k¼1. On Sj0 we have u~ ðx, yÞ z0 ðx, yÞ ¼ ð y j0 Þðuðx, yÞ z0 ðx, yÞÞ and thus we obtain Dðu~ , z0 Þj0 Dðu, z0 Þj0 from which ’j0 ðu~ Þ ~ . Therefore m ðu~ Þ ðuÞ . j0 X k¼1. ’k ðuÞ þ ~.

(59) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1562. ALESSIO, JEANJEAN, AND MONTECCHIARI. P0 from which, since ðuÞ m þ ~ we conclude that j1 ’k ðuÞ 2~ : In particular ’j ðuÞ 2~ for any j j0 and thus Dðu, z0 Þj ~ for any j j0 . This proves ðiiÞ. In a similar way one proves ðiiiÞ. To prove the last part of the lemma we argue by contradiction assuming that there exist j0 2 Z and 2 Znf0, g such that Dðu, z Þj0 . Let u1 ¼ þ and u2 ¼ and note that u1 2 H while , j0 ðuÞ , j0 ðuÞ u2 ð , Þ 2 H . Thus, by the choice of , we obtain ðu1 Þ þ ðu2 Þ 2m . Now, we have ðu1 Þ þ ðu2 Þ ðuÞ þ ’j0 ðu1 Þ þ ’j0 ðu2 Þ m þ ~ þ ’j0 ðu1 Þ þ ’j0 ðu2 Þ but since Dðu, z0 Þj0 , arguing as above, we obtain ’j0 ðu1 Þ þ ’j0 ðu2 Þ 2~ , which leads to the contradiction 3~ m . œ According to Lemma 3.8, we fix ~ 2 ð0, ~0 Þ such that ~ ¼ ~ ~ < =4 and we will denote ~ ¼ ~ ~ and ‘~ ¼ ‘~ ~. We are now able to prove the following compactness property of the minimizing sequences of in H . They will be sufficient to use the direct method of the Calculus of Variation to show that the functional admits a minimum in the class H as stated in Theorem 3.1 below. Lemma 3.9. Let ðun Þ H and > 0 be such that ðun Þ ! m and Dðun , KÞ0 for any n 2 N. Then there exists u 2 K such that, up to a 1 subsequence, un ! u as n ! 1 weakly in Hloc ðR2 Þ. Proof. Without loss of generality we can assume that ~ and that ðun Þ m þ ~ for any n 2 N. Then < ~ =4 and, since Dðun , KÞ0 , by Lemma 3.8 we obtain that for any n 2 N, Dðun , z0 Þj . , 8 j ‘~ 4. and. Dðun , z Þj . , 8 j ‘~ : 4. ð3:2Þ. By Lemma 3.5 there exists u 2 such that along a subsequence un ! u 1 weakly in Hloc ðR2 Þ. Finally, by (3.2) and the weakly lower semicontinuity of the distance we obtain Dðu, z0 Þj . , 8 j ‘~ 4. and. Dðu, z Þj . , 8 j ‘~ : 4. ð3:3Þ. By Lemma 3.6, we conclude that Dðu, z0 Þj ! 0 as j ! 1 and Dðu, z Þ ! 0 as j ! þ1, that is u 2 H . Then, by semicontinuity, ðuÞ ¼ m and the lemma follows. œ.

(60) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. PERIODIC ALLEN–CAHN EQUATIONS. 1563. Theorem 3.1. If ðÞ holds there exists u 2 K . Moreover, u 2 C2 ðR2 Þ and is a classical solution of (P) with kukL1 ðR2 Þ R0 . Proof. Let ðun Þ H be such that ðun Þ ! m . By ðH2 Þ-ðiiÞ we can assume that kun kL1 ðR2 Þ R0 , indeed otherwise one can consider the minimizing sequence u~ n ¼ maxfminfun , R0 g, R0 g. Moreover since, for any n 2 N, Dðun , z0 Þj ! 0 as j ! 1 and lim inf j!þ1 Dðun , z0 Þj Dðz0 , z Þ0 , by Lemma 3.3, there exists kn 2 Z such that ’kn ðun Þ =2. Then, by ðÞ, there exists > 0 such that Distðun , KÞkn for any n 2 N. Setting vn ¼ un ð, þkn Þ, we have vn 2 H , ðvn Þ ¼ ðun Þ and Distðvn , KÞ0 . By Lemma 3.9 there exists u 2 K such that along a 1 subsequence vn ! u weakly in Hloc ðR2 Þ and by semicontinuity kukL1 ðR2 Þ R0 . Standard elliptic arguments implies that u 2 C2 ðR2 Þ and is a classical solution of (E). Moreover, since kukL1 ðR2 Þ R0 and Dðu, z0 Þj ! 0 as j ! 1 and Dðu, z Þ ! 0 as j ! þ1, we can prove, as in Proposition 2.1 and in Ref. [5], that u satisfies the asymptotic conditions in the x variable uniformly with respect to y 2 R. œ As last step of this section we further characterize the compactness properties of the minimizing sequences of in H . Lemma 3.10 below strengthens the result obtained in Lemma 3.9 and will be used in the next section. Lemma 3.10. Let ðun Þ H and > 0 be such that ðun Þ ! m and Dðun , KÞ0 for any n 2 N. Then there exists u 2 K such that, up to a subsequence, Dðun , uÞj ! 0 as n ! 1, for all j 2 Z. Proof. By Lemma 3.9 there exists a subsequence of ðun Þ, still denoted ðun Þ, 1 and u 2 K such that un ! u weakly in Hloc ðR2 Þ. We claim that ’j ðun Þ ! ’j ðuÞ for all j 2 Z. Indeed by semicontinuity, ’j ðuÞ lim inf ’j ðun Þ for all j 2 Z. By contradiction, assume that there exists j0 2 Z such that lim supð’j0 ðun Þ ’j0 ðuÞÞ ¼ " > 0 and so that there exists a subsequence ðuni Þ of ðun Þ such that ’j0 ðuni Þ ’j0 ðuÞ ! " as i ! 1. Since X j jjk. ð’j ðun Þ ’j ðuÞÞ . X. ’j ðuÞ ! 0,. as k ! 1. j jjk. P we can fix k > j j0 j such P that j jjk ð’j ðun Þ ’j ðuÞÞ ð"=2Þ. Then, setting bn ¼ ðun Þ ðuÞ ¼ j2Z ð’j ðun Þ ’j ðuÞÞ, since ðun Þ ! m ¼ ðuÞ, we.

(61) Downloaded By: [EBSCOHost EJS Content Distribution] At: 15:28 21 July 2008. MARCEL DEKKER, INC. • 270 MADISON AVENUE • NEW YORK, NY 10016 ©2002 Marcel Dekker, Inc. All rights reserved. This material may not be used or reproduced in any form without the express written permission of Marcel Dekker, Inc.. 1564. ALESSIO, JEANJEAN, AND MONTECCHIARI. have bn ! 0 as n ! 1, while X. bni ¼ ’j0 ðuni Þ ’j0 ðuÞ þ. ð’j ðuni Þ ’j ðuÞÞ þ. j jj<k, j6¼j0. X. ’j0 ðuni Þ ’j0 ðuÞ þ. j jj<k, j6¼j0. X. ð’j ðuni Þ ’j ðuÞÞ. j jjk. " ð’j ðuni Þ ’j ðuÞÞ : 2. Since lim infð’j ðun Þ ’j ðuÞÞ 0 for all j 2 Z, we obtain the contradiction lim inf bni "=2. In the sequel we will consider only the strip S0 , the proof being the same for the other strips. Recalling that ’ðuÞ ¼ ’0 ðuÞ þ c, we have already proved that ’ðun Þ ! ’ðuÞ. By Lemma 2.1 we obtain that Z. Z jrun j2 dx dy ! jruj2 dx dy and S0 S0 Z Z aðx, yÞWðun Þ dx dy ! aðx, yÞWðuÞ dx dy: S0. ð3:4Þ. S0. Since ðrun Þ is a bounded sequence in L2 ðS0 Þ, we deduce that run ! ru 1 weakly in L2 ðS0 Þ (recall that un ! u weakly in Hloc ðR2 Þ). Then, since by (3.4) krun kL2 ðS0 Þ ! krukL2 ðS0 Þ , we conclude run ! ru in L2 ðS0 Þ. as n ! 1:. ð3:5Þ. By (3.5), to prove the lemma it remains to show that un u ! 0 in L2 ðS0 Þ. For that, since un u ! 0 in L2loc ðR2 Þ, it suffices that for any " > 0 there exist nð"Þ 2 N and L" > 0 such that Z. Z. 1. jun uj2 dy dx ". for any n nð"Þ:. 0. jxj>L". We shall prove that for any " > 0 there exist nð"Þ 2 N and L" > 0 such that Z. Z. 1. 2. Z. Z. 1. ju þ bj dy dx þ Z. x<L". x<L". Z. 0. x>L". 1. jun þ bj2 dy dx þ. Z. 0. and the proof will be complete.. x>L". ju bj2 dy dx ",. ð3:6Þ. 0. Z. 1 0. jun bj2 dy dx " 8 n nð"Þ ð3:7Þ.