a singularly perturbed dirichlet problem for the laplace operator in a periodically perforated...

Research Article

Received 31 January 2011 Published online 30 December 2011 in Wiley Online Library

(wileyonlinelibrary.com) DOI: 10.1002/mma.1575

A singularly perturbed Dirichlet problem forthe Laplace operator in a periodicallyperforated domain. A functionalanalytic approach

Paolo Musolino*†

Communicated by A. Kirsch

Let� be a sufficiently regular bounded connected open subset of Rn such that 0 2� and that Rn n cl� is connected. Thenwe take q11, : : : , qnn 2�0,C1Œ and p 2 Q �

QnjD1�0, qjjŒ. If � is a small positive number, then we define the periodically per-

forated domain SŒ�� Rn n [z2Zn cl�

pC ��CPn

jD1.qjjzj/ej

�, where fe1, : : : , eng is the canonical basis of Rn. For � small

and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set SŒ��. Namely, we considera Dirichlet condition on the boundary of the set pC ��, together with a periodicity condition. Then we show real analyticcontinuation properties of the solution and of the corresponding energy integral as functionals of the pair of � and of theDirichlet datum on pC �@�, around a degenerate pair with � D 0. Copyright © 2011 John Wiley & Sons, Ltd.

Keywords: Boundary value problems for second-order elliptic equations; integral representations, integral operators, integral equa-tions methods; singularly perturbed domain; Laplace operator; periodically perforated domain; real analytic continuationin Banach space

1. Introduction

In this article, we consider a Dirichlet problem in a periodically perforated domain with small holes. We fix once for all a natural number

n 2N n f0, 1g

and

.q11, : : : , qnn/ 2�0,C1Œn

and a periodicity cell

Q�…njD1�0, qjjŒ .

Then we denote by q the diagonal matrix

q�

0BB@q11 0 : : : 0

0 q22 : : : 0: : : : : : : : : : : :

0 0 : : : qnn

1CCA ,

by jQj the measure of the fundamental cell Q, and by �Q the outward unit normal to @Q, where it exists. Clearly,

qZn � fqz : z 2 Zng

Dipartimento di Matematica Pura ed Applicata, Università di Padova , Via Trieste 63, 35121 Padova, Italy*Correspondence to: Paolo Musolino, Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste 63, 35121 Padova, Italy.†E-mail: [email protected]

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Copyright © 2011 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2012, 35 334–349

P. MUSOLINO

is the set of vertices of a periodic subdivision of Rn corresponding to the fundamental cell Q. Let

m 2N n f0g , ˛ 2�0, 1Œ .

Then we take a point p 2 Q and a bounded connected open subset � of Rn of class Cm,˛ such that �� Rn n cl� is connected andthat 0 2�. If � 2R, then we set

�� pC �� .

Then we take �0 > 0 such that cl�� Q for j�j< �0, and we introduce the periodically perforated domain

SŒ�� Rn n [z2Zn cl.�� C qz/ ,

for � 2�� 0, �0Œ. Next, we fix a function g0 2 Cm,˛.@�/. For each pair .�, g/ 2�0, �0Œ�Cm,˛.@�/, we consider the Dirichlet problem8<:�u.x/D 0 8x 2 SŒ�� ,u.xC qei/D u.x/ 8x 2 clSŒ�� , 8i 2 f1, : : : , ngu.x/D g

� 1� .x � p/

�8x 2 @�� ,

(1.1)

where fe1, : : : , eng is the canonical basis of Rn. If .�, g/ 2�0, �0Œ�Cm,˛.@�/, then problem (1.1) has a unique solution in Cm,˛.clSŒ�� /,and we denote it by uŒ�, g�.�/ (cf. Proposition 2.10.)

Then we pose the following questions:

(i) Let x be fixed in Rn n .pC qZn/. What can be said on the map .�, g/ 7! uŒ�, g�.x/ around .�, g/D .0, g0/?(ii) What can be said on the map .�, g/ 7!

RQncl��

jDxuŒ�, g�.x/j2 dx around .�, g/D .0, g0/?

Questions of this type have long been investigated, for example, for problems on a bounded domain with a small hole with themethods of asymptotic analysis, which aim at giving complete asymptotic expansions of the solutions in terms of the parameter �. Itis perhaps difficult to provide a complete list of the contributions. Here, we mention the work of Ammari and Kang [1, Ch. 5], Ammari,Kang, and Lee [2, Ch. 3], Kozlov, Maz’ya, and Movchan [3], Maz’ya, Nazarov, and Plamenewskij [4, 5], Ozawa [6], Vogelius and Volkov [7],Ward and Keller [8]. We also mention the vast literature of homogenization theory (cf., e.g., Dal Maso and Murat [9].)

Here instead, we wish to characterize the behavior of uŒ�, g�.�/ at .�, g/ D .0, g0/ by a different approach. Thus for example, if weconsider a certain functional, say f .�, g/, relative to the solution such as, for example, one of those considered in questions (i)–(ii), wewould try to prove that f .�, �/ can be continued real analytically around .�, g/D .0, g0/. We observe that our approach does have certainadvantages (cf., e.g., Lanza [10].) Such a project has been carried out by Lanza de Cristoforis in several papers for problems in a boundeddomain with a small hole (cf., e.g., Lanza [11–14].) In the frame of linearized elastostatics, we also mention, for example, Dalla Riva andLanza [15, 16].

As far as problems in periodically perforated domains are concerned, we mention, for instance, the work of Ammari, Kang, and Touibi[17], where a linear transmission problem is considered in order to compute an asymptotic expansion of the effective electrical conduc-tivity of a periodic dilute composite (see also Ammari and Kang [1, Ch. 8].) Furthermore, we note that periodically perforated domainsare extensively studied in the frame of homogenization theory. Among the vast literature, here, we mention, for example, Cioranescuand Murat [18, 19], Ansini and Braides [20]. We also observe that boundary value problems in domains with periodic inclusions can beanalyzed, at least for the two dimensional case, with the method of functional equations. Here, we mention, for example, Mityushevand Adler [21], Rogosin, Dubatovskaya, and Pesetskaya [22], Castro and Pesetskaya [23].

We now briefly outline our strategy. We first convert problem (1.1) into an integral equation by exploiting potential theory. Thenwe observe that the corresponding integral equation can be written, after an appropriate rescaling, in a form that can be analyzed bymeans of the Implicit Function Theorem around the degenerate case in which .�, g/ D .0, g0/, and we represent the unknowns of theintegral equation in terms of � and g. Next, we exploit the integral representation of the solutions, and we deduce the representationof uŒ�, g�.�/ in terms of � and g.

This article is organized as follows. Section 2 is a section of preliminaries. In Section 3, we formulate problem (1.1) in terms of anintegral equation, and we show that the solutions of the integral equation depend real analytically on � and g. In Section 4, we showthat the results of Section 3 can be exploited to prove our main Theorem 4.1 on the representation of uŒ�, g�.�/, and Theorem 4.6 onthe representation of the energy integral of uŒ�, g�.�/ on a perforated cell. At the end of this article, we have enclosed an Appendix withsome results exploited in the paper.

2. Preliminaries and notation

We now introduce the notation in accordance with Lanza [13, p. 66].We denote the norm on a normed space X by k � kX . Let X and Y be normed spaces. We endow the space X � Y with the norm

defined by k.x, y/kX�Y � kxkX C kykY for all .x, y/ 2 X � Y , whereas we use the Euclidean norm for Rn. For standard definitions ofCalculus in normed spaces, we refer to Prodi and Ambrosetti [24]. The symbol N denotes the set of natural numbers including 0. Theinverse function of an invertible function f is denoted f .�1/, as opposed to the reciprocal of a real-valued function g or the inverse ofa matrix A, which are denoted g�1 and A�1, respectively. A dot ‘�’ denotes the inner product in Rn. Let A be a matrix. Then At denotes

the transpose matrix of A and Aij denotes the .i, j/-entry of A. If A is invertible, we set A�t ��

A�1�t

. Let D � Rn. Then clD denotes the


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P. MUSOLINO

closure of D and @D denotes the boundary of D. For all R > 0, x 2Rn, xj denotes the jth coordinate of x, jxj denotes the Euclidean mod-ulus of x in Rn, and Bn.x, R/ denotes the ball fy 2Rn : jx� yj< Rg. The symbol idn denote the identity map from Rn, that is, idn.x/D xfor all x 2 Rn. If z 2 C, then z denotes the conjugate complex number of z. Let � be an open subset of Rn. The space of m timescontinuously differentiable real-valued functions on� is denoted by Cm.�,R/, or more simply by Cm.�/. D.�/ denotes the space offunctions of C1.�/ with compact support. The dual D0.�/ denotes the space of distributions in �. Let r 2 N n f0g. Let f 2 .Cm.�//r .

The sth component of f is denoted fs, and Df denotes the Jacobian matrix�@fs@xl

�sD1,:::,r,lD1,:::,n

. Let �� .�1, : : : , �n/ 2Nn, j�j � �1 C � � � C �n.

Then D� f denotes @j�jf@x�11 :::@x�n

n. The subspace of Cm.�/ of those functions f whose derivatives D� f of order j�j6m can be extended with

continuity to cl� is denoted Cm.cl�/. The subspace of Cm.cl�/ whose functions have mth order derivatives that are Hölder contin-uous with exponent ˛ 2�0, 1� is denoted Cm,˛.cl�/ (cf., e.g., Gilbarg and Trudinger [25].) The subspace of Cm.cl�/ of those functionsf such that fjcl.�\Bn.0,R// 2 Cm,˛.cl.� \ Bn.0, R/// for all R 2�0,C1Œ is denoted Cm,˛

loc .cl�/. Let D � Rr . Then Cm,˛.cl�,D/ denotes˚f 2 .Cm,˛.cl�//r : f .cl�/�D

�.

Now let � be a bounded open subset of Rn. Then Cm.cl�/ and Cm,˛.cl�/ are endowed with their usual norm and are well knownto be Banach spaces (cf., e.g., Troianiello [26, §1.2.1].) We say that a bounded open subset � of Rn is of class Cm or of class Cm,˛ , if itsclosure is a manifold with boundary imbedded in Rn of class Cm or Cm,˛ , respectively (cf., e.g., Gilbarg and Trudinger [25, §6.2].) Wedenote by �� the outward unit normal to @�. For standard properties of functions in Schauder spaces, we refer the reader to Gilbargand Trudinger [25] and to Troianiello [26] (see also Lanza [27, §2, Lem. 3.1, 4.26, Thm. 4.28], Lanza and Rossi [28, §2].)

We retain the standard notation of Lp spaces and of corresponding norms.If M is a manifold imbedded in Rn of class Cm,˛ , with m > 1, ˛ 2�0, 1Œ, one can define the Schauder spaces also on M by exploit-

ing the local parametrizations. In particular, one can consider the spaces Ck,˛.@�/ on @� for 0 6 k 6 m with � a bounded openset of class Cm,˛ , and the trace operator from Ck,˛.cl�/ to Ck,˛.@�/ is linear and continuous. Moreover, for each R > 0 such thatcl�� Bn.0, R/, there exists a linear and continuous extension operator from Ck,˛.@�/ to Ck,˛.cl�/, and of Ck,˛.cl�/ to Ck,˛.clBn.0, R//(cf., e.g., Troianiello [26, Thm. 1.3, Lem. 1.5].) We denote by d� the area element of a manifold imbedded in Rn.

We note that throughout the paper, ‘analytic’ means ‘real analytic’. For the definition and properties of analytic operators, we referto Prodi and Ambrosetti [24, p. 89] and to Deimling [29, p. 150]. Here we just recall that if X , Y are (real) Banach spaces, and if F is anoperator from an open subset W of X to Y , then F is real analytic in W if for every x0 2W there exist r > 0 and continuous symmetricn-linear operators An from X n to Y such that

Pn>1 kAnkrn < 1 and F.x0 C h/ D F.x0/C

Pn>1 An.h, : : : , h/ for khkX 6 r (cf., e.g.,

Prodi and Ambrosetti [24, p. 89] and Deimling [29, p. 150].) In particular, we mention that the pointwise product in Schauder spacesis bilinear and continuous, and thus analytic, and that the map, which takes a nonvanishing function to its reciprocal, or an invertiblematrix of functions to its inverse matrix is real analytic in Schauder spaces (cf., e.g., Lanza and Rossi [28, pp. 141, 142].)

We denote by Sn the function from Rn n f0g to R defined by

Sn.x/�

(1sn

log jxj 8x 2Rn n f0g, if nD 2 ,1

.2�n/snjxj2�n 8x 2Rn n f0g, if n > 2 ,

where sn denotes the .n� 1/-dimensional measure of @Bn. Sn is well-known to be the fundamental solution of the Laplace operator.If y 2Rn and f is a function defined in Rn, we set yf .x/� f .x � y/ for all x 2Rn. If u is a distribution in Rn, then we set

< yu, f >D< u, �yf > 8f 2D.Rn/ .

We denote by E2� iq�1z , the function defined by

E2� iq�1z.x/� e2� i.q�1z/�x 8x 2Rn ,

for all z 2 Zn.If� is an open subset of Rn, k 2N , ˇ 2�0, 1�, we set

Ckb.cl�/� fu 2 Ck.cl�/ : D�u is bounded 8 2Nn such that j j6 kg ,

and we endow Ckb.cl�/with its usual norm

kukCkb.cl�/ �

Xj� j6k

supx2cl�

jD�u.x/j 8u 2 Ckb.cl�/ .

Then we set

Ck,ˇb .cl�/� fu 2 Ck,ˇ .cl�/ : D�u is bounded 8 2Nn such that j j6 kg ,

and we endow Ck,ˇb .cl�/with its usual norm

kukCk,ˇ

b .cl�/�Xj� j6k

supx2cl�

jD�u.x/j CXj� jDk

jD�u : cl�jˇ 8u 2 Ck,ˇb .cl�/ ,

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P. MUSOLINO

where jD�u : cl�jˇ denotes the ˇ-Hölder constant of D�u.Next we turn to periodic domains. If I is an arbitrary subset of Rn such that clI � Q, then we set

SŒI��[

z2Zn

.qzC I/D qZnC I ,

SŒI�� Rn n clSŒI� .

We note that if Rn n clI is connected, then SŒI�� is connected.Let D �Rn be such that qzCD �D for all z 2 Zn. We say that a function u from D to R is q-periodic if u.xCqej/D u.x/ for all x 2D

and for all j 2 f1, : : : , ng.If I is an open subset of Rn such that clI � Q and if k 2N , ˇ 2�0, 1�, then we set

Ckq.clSŒI�/�

nu 2 Ck.clSŒI�/ : u is q� periodic

o,

which we regard as a Banach subspace of Ckb.clSŒI�/, and

Ck,ˇq .clSŒI�/�

nu 2 Ck,ˇ .clSŒI�/ : u is q� periodic

o,

which we regard as a Banach subspace of Ck,ˇb .clSŒI�/, and

Ckq.clSŒI��/�

nu 2 Ck.clSŒI��/ : u is q� periodic

o,

which we regard as a Banach subspace of Ckb.clSŒI��/, and

Ck,ˇq .clSŒI��/�

nu 2 Ck,ˇ .clSŒI��/ : u is q� periodic

o,

which we regard as a Banach subspace of Ck,ˇb .clSŒI��/. We denote by S.Rn/ the Schwartz space of complex-valued rapidly

decreasing functions.In the following Theorem, we introduce a periodic analog of the fundamental solution of the Laplace operator (cf., e.g., Hasimoto

[30], Shcherbina [31], Poulton, Botten, McPhedran, and Movchan [32], Ammari, Kang, and Touibi [17], Ammari and Kang [1, p. 53],and [33].)

Theorem 2.1The generalized series

Sqn �

Xz2Znnf0g

1

�4�2jq�1zj2jQjE2� iq�1z (2.2)

defines a tempered distribution in Rn such that Sqn is q-periodic, that is,

qjj ej Sqn D Sq

n 8j 2 f1, : : : , ng ,

and such that

�Sqn D

Xz2Zn

ıqz �1

jQj,

where ıqz denotes the Dirac measure with mass at qz, for all z 2 Zn. Moreover, the following statements hold.

(i) Sqn is real analytic in Rn n qZn.

(ii) Rqn � Sq

n � Sn is real analytic in .Rn n qZn/[ f0g, and we have

�Rqn D

Xz2Znnf0g

ıqz �1

jQj,

in the sense of distributions.(iii) Sq

n 2 L1loc.R

n/.

(iv) Sqn.x/D Sq

n.�x/ for all x 2Rn n qZn.

Proof. For the proof of (i), (ii), we refer for example to [33], where an analog of a periodic fundamental solution for a second orderstrongly elliptic differential operator with constant coefficients has been constructed. We now consider statement (iii). As is well known,Sq

n is a locally integrable complex-valued function (cf. [33, §3].) By the definition of Sqn, and by the equality

< E2� iq�1z , >D< E2� iq�1.�z/, > 8 2 S.Rn/ , 8z 2 Zn n f0g ,


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P. MUSOLINO

and by the obvious identity

1

�4�2j � q�1zj2jQjD

1

�4�2jq�1zj2jQj8z 2 Zn n f0g ,

we can conclude that Sqn is actually a real-valued function. We now turn to the proof of (iv). By a straightforward verification based on

equality (2.2), we have ZRn

Sqn.x/ .�x/dx D

ZRn

Sqn.x/ .x/dx 8 2 S.Rn/ ,

and thus Sqn.x/D Sq

n.�x/ for all x 2Rn n qZn. Hence, the proof is complete �

We now introduce the periodic double layer potential. Let ˛ 2�0, 1Œ, m 2 N n f0g. Let I be a bounded connected open subset of Rn

of class Cm,˛ such that Rn n clI is connected and that clI � Q. Let Sqn be as in Theorem 2.1. If � 2 C0,˛.@I/, we set

wqŒ@I,��.x/��

[email protected]

n.x � y//�I.y/�.y/d�y 8x 2Rn .

In the following Theorem, we collect some properties of the periodic double layer potential.

Theorem 2.3Let ˛ 2�0, 1Œ, m 2 N n f0g. Let I be a bounded connected open subset of Rn of class Cm,˛ such that Rn n clI is connected and thatclI � Q. Let Sq

n be as in Theorem 2.1. Then the following statements hold.

(i) Let � 2 C0,˛.@I/. Then wqŒ@I,�� is q-periodic and

�.wqŒ@I,��/.x/D 0 8x 2Rn n @SŒI� .

(ii) If � 2 Cm,˛.@I/, then the restriction wqŒ@I,��jSŒI� can be extended uniquely to an element wCq Œ@I,�� of Cm,˛q .clSŒI�/, and the

restriction wqŒ@I,��jSŒI�� can be extended uniquely to an element w�q Œ@I,�� of Cm,˛q .clSŒI��/, and we have

w˙q Œ@I,��D˙1

2�CwqŒ@I,�� on @I , (2.4)

.DwCq Œ@I,��/�I � .Dw�q Œ@I,��/�I D 0 on @I . (2.5)

(iii) The operator from Cm,˛.@I/ to Cm,˛q .clSŒI�/, which takes � to the function wCq Œ@I,��, is continuous. The operator from Cm,˛.@I/

to Cm,˛q .clSŒI��/which takes � to the function w�q Œ@I,�� is continuous.

(iv) The following equalities hold

wqŒ@I, 1�.x/D1

2�jIj

jQj8x 2 @SŒI� , (2.6)

wqŒ@I, 1�.x/D 1�jIj

jQj8x 2 SŒI� , (2.7)

wqŒ@I, 1�.x/D�jIj

jQj8x 2 SŒI�� , (2.8)

where jIj, jQj denote the n-dimensional measure of I and of Q, respectively.

Proof. For the proof of statements (i), (ii), (iii), we refer, for example, to [33]. We now consider statement (iv). It clearly suffices to proveequality (2.8). Indeed, equalities (2.6) and (2.7) can be proved by exploiting equality (2.8) and the jump relations of equality (2.4). By theperiodicity of wqŒ@I, 1�, we can assume x 2 clQ n clI. By the Green formula and Theorem 2.1, we have

�

[email protected]

n.x � y//�I.y/d�y D

ZI�y.S

qn.x � y//dy D�

jIj

jQj,

and accordingly, equality (2.8) holds. Thus the proof is complete. �

Let ˛ 2�0, 1Œ, m 2N n f0g. If˝ is a bounded connected open subset of Rn of class Cm,˛ , we find convenient to set

Cm,˛.@˝/0 �

�f 2 Cm,˛.@˝/W

Z@˝

f d� D 0

�.

Then we have the following Proposition.33

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P. MUSOLINO

Proposition 2.9Let ˛ 2�0, 1Œ, m 2 N n f0g. Let I be a bounded connected open subset of Rn of class Cm,˛ such that Rn n clI is connected and thatclI � Q. Let Sq

n be as in Theorem 2.1. Let MŒ�, �� be the map from Cm,˛.@I/0 �R to Cm,˛.@I/, defined by

MŒ�, ��.x/��1

2�.x/CwqŒ@I,��.x/C � 8x 2 @I ,

for all .�, �/ 2 Cm,˛.@I/0 �R. Then MŒ�, �� is a linear homeomorphism from Cm,˛.@I/0 �R onto Cm,˛.@I/.

Proof. By Theorem 2.3, M is continuous. As a consequence, by the Open Mapping Theorem, it suffices to prove that M is a bijection.We first show that M is injective. So let .�, �/ 2 Cm,˛.@I/0 �R be such that MŒ�, ��D 0. Then,

�1

2�.x/CwqŒ@I,��.x/D�� 8x 2 @I ,

and thus, by Proposition A.3 of the Appendix, � must be constant. BecauseR@I � d� D 0, then � D 0, and so also � D 0. Hence, M is

injective. It remains to prove that M is surjective. So let g 2 Cm,˛.@I/. We need to prove that there exists a pair .�, �/ 2 Cm,˛.@I/0 �Rsuch that MŒ�, ��D g. By Proposition A.3 of the Appendix, there exists a (unique) Q� 2 Cm,˛.@I/ such that

�1

2Q�.x/CwqŒ@I, Q��.x/D g.x/ 8x 2 @I .

Accordingly, if we set

�.x/� Q�.x/�1R

@I d�

Z@IQ� d� 8x 2 @I ,

� ��jIj

jQj

1R@I d�

Z@IQ� d� ,

where jIj, jQj denote the n-dimensional measure of I and of Q, respectively, then clearly .�, �/ 2 Cm,˛.@I/0 � R and MŒ�, �� D g.Therefore, M is bijective, and the proof is complete. �

In the following Proposition, we show that a periodic Dirichlet boundary value problem in the perforated domain SŒI�� has a uniquesolution in Cm,˛

q .clSŒI��/, which can be represented as the sum of a periodic double layer potential and a constant.

Proposition 2.10Let ˛ 2�0, 1Œ, m 2 N n f0g. Let I be a bounded connected open subset of Rn of class Cm,˛ such that Rn n clI is connected and thatclI � Q. Let Sq

n be as in Theorem 2.1. Let � 2 Cm,˛.@I/. Then the following boundary value problem8<:�u.x/D 0 8x 2 SŒI�� ,u.xC qei/D u.x/ 8x 2 clSŒI�� , 8i 2 f1, : : : , ng ,u.x/D � .x/ 8x 2 @I ,

(2.11)

has a unique solution u 2 Cm,˛q .clSŒI��/. Moreover,

u.x/D w�q Œ@I,��.x/C � 8x 2 clSŒI�� , (2.12)

where .�, �/ is the unique solution in Cm,˛.@I/0 �R of the following integral equation

� .x/D�1

2�.x/CwqŒ@I,��.x/C � 8x 2 @I . (2.13)

Proof. We first note that Proposition A.1 of the Appendix implies that problem (2.11) has at most one solution. As a consequence,we need to prove that the function defined by equality (2.12) solves problem (2.11). By Proposition 2.9, there exists a unique solution.�, �/ 2 Cm,˛.@I/0�R of Equation (2.13). Then, by Theorem 2.3 and Equation (2.13), the function defined by equality (2.12) is a periodicharmonic function satisfying the third condition of problem (2.11), and thus a solution of problem (2.11). �

Remark 2.14Let the assumptions of Proposition 2.10 hold. We note that we proved, in particular, that the solution of boundary value problem (2.11)can be represented as the sum of a periodic double layer potential and a constant. However, we observe that we could also representthe solution of problem (2.11) as a periodic double layer potential (cf. Proposition A.3 of the Appendix.) On the other hand, for theanalysis of problem (1.1) around the degenerate value .�, g/ D .0, g0/, it will be preferable to exploit the representation formula ofProposition 2.10.


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P. MUSOLINO

3. Formulation of the problem in terms of integral equations

We now provide a formulation of problem (1.1) in terms of an integral equation. We shall consider the following assumptions for some˛ 2�0, 1Œ and for some natural m> 1.

Let˝ be a bounded connected open subset of Rn of class Cm,˛ such that Rn n cl˝ is connected and that 0 2˝ .

Let p 2 Q. (3.1)

If � 2R, we set

˝� � pC �˝ .

Now let

�0 � supn� 2�0,C1ŒW cl˝� � Q ,8� 2�� , �Œ

o. (3.2)

A simple topological argument shows that if assumption (3.1) holds, then SŒ˝� �� is connected, for all � 2��0, �0Œ. We also note that

�˝� .pC �t/D sgn.�/�˝.t/ 8t 2 @˝ ,

for all � 2�� 0, �0Œnf0g, where sgn.�/D 1 if � > 0, sgn.�/D�1 if � < 0. Then we shall consider the following assumption.

Let g0 2 Cm,˛.@˝/. (3.3)

If .�, g/ 2�0, �0Œ�Cm,˛.@˝/, we shall convert our boundary value problem (1.1) into an integral equation. We could exploit Proposi-tion 2.10, with I replaced by˝� , but we note that the integral equation and the corresponding integral representation of the solutioninclude integrations on the �-dependent domain @˝� . To get rid of such a dependence, we shall introduce the following Lemma, inwhich we properly rescale the unknown density.

Lemma 3.4Let ˛ 2�0, 1Œ. Let m 2N n f0g. Let assumptions (3.1)–(3.3) hold. Let Sq

n, Rqn be as in Theorem 2.1. Let .�, g/ 2�0, �0Œ�Cm,˛.@˝/. Then a pair

.� , �/ 2 Cm,˛.@˝/0 �R solves equation

�1

2�.t/�

Z@˝.DSn.t� s//�˝.s/�.s/d�s � �

n�1Z@˝.DRq

n.�.t� s///�˝.s/�.s/d�sC � D g.t/ 8t 2 @˝ , (3.5)

if and only if the pair .�, �/ 2 Cm,˛.@˝�/0 �R, with � delivered by

�.x/� ��1

�.x � p/

�8x 2 @˝� , (3.6)

solves equation

�1

2�.x/CwqŒ@˝� ,��.x/C � D � .x/ 8x 2 @˝� , (3.7)

where

� .x/� g�1

�.x � p/

�8x 2 @˝� .

Moreover, Equation (3.5) has a unique solution in Cm,˛.@˝/0 �R.

Proof. The equivalence of Equation (3.5) in the unknown .� , �/ and Equation (3.7) in the unknown .�, �/, with � delivered byequality (3.6), is a straightforward consequence of the Theorem of change of variables in integrals. Then the existence and unique-ness of a solution in Cm,˛.@˝/0 �R of Equation (3.5), follows from Proposition 2.9 applied to Equation (3.7), and from the equivalenceof Equations (3.5) and (3.7). �

In the following Lemma, we study Equation (3.5), when .�, g/D .0, g0/.

Lemma 3.8Let ˛ 2�0, 1Œ. Let m 2N n f0g. Let assumptions (3.1)–(3.3) hold. Let 0 be the unique solution in Cm�1,˛.@˝/ of the following problem�

� 12 .t/C

R@˝.DSn.t� s//�˝.t/.s/d�s D 0 8t 2 @˝ ,R

@˝ d� D 1 .(3.9)

Then equation

�1

2�.t/�

Z@˝.DSn.t� s//�˝.s/�.s/d�sC � D g0.t/ 8t 2 @˝ , (3.10)3

40


P. MUSOLINO

which we call the limiting equation, has a unique solution in Cm,˛.@˝/0 �R, which we denote by . Q� , Q�/. Moreover,

Q� D

Z@˝

g00 d� , (3.11)

and the function Qu 2 Cm,˛.Rn n˝/, defined by

Qu.t/��

Z@˝.DSn.t� s//�˝.s/ Q�.s/d�s 8t 2Rn n cl˝ , (3.12)

and extended by continuity to Rn n˝ , is the unique solution in Cm,˛.Rn n˝/ of the following problem8<:�u.t/D 0 8t 2Rn n cl˝ ,u.t/D g0.t/�

R@˝ g00 d� 8t 2 @˝ ,

limt!1 u.t/D 0 .(3.13)

Proof. We first note that the unique solvability of problem (3.9) in the class of continuous functions follows by classical potentialtheory (cf., e.g., Folland [34, Ch. 3].) For the Cm�1,˛ regularity of the solution, we refer, for example, to Lanza [10, Appendix A]. By Propo-sition A.5 of the Appendix, Equation (3.10) has a unique solution in Cm,˛.@˝/0 � R. Moreover, as is well known, if 2 Cm,˛.@˝/,then

2n�

1

2�.�/�

Z@˝.DSn.� � s//�˝.s/�.s/d�sW � 2 Cm,˛.@˝/

oif and only if Z

@˝ 0 d� D 0 ,

and thus Q� must be delivered by equality (3.11) (cf., e.g., Folland [34, Ch. 3] and Lanza [10, Appendix A].) Then by classical potentialtheory, the function defined by equality (3.12) and extended by continuity to Rn n ˝ solves problem (3.13), which has at most onesolution (cf., e.g., Folland [34, Ch. 3], Miranda [35], Dalla Riva and Lanza [36, Theorem 3.1], Lanza and Rossi [28, Theorem 3.1].) �

We are now ready to analyze Equation (3.5) around the degenerate case .�, g/ D .0, g0/. We find convenient to introduce thefollowing abbreviation. We set

Xm,˛ � Cm,˛.@˝/0 �R .

Then we have the following.

Proposition 3.14Let ˛ 2�0, 1Œ. Let m 2 N n f0g. Let assumptions (3.1)–(3.3) hold. Let Sq

n, Rqn be as in Theorem 2.1. Let � be the map from

�� 0, �0Œ�Cm,˛.@˝/�Xm,˛ to Cm,˛.@˝/, defined by

�Œ�, g, � , ��.t/��1

2�.t/�

Z@˝.DSn.t� s//�˝.s/�.s/d�s � �

n�1Z@˝.DRq

n.�.t� s///�˝.s/�.s/d�sC � � g.t/ 8t 2 @˝ ,

for all .�, g, � , �/ 2�� 0, �0Œ�Cm,˛.@˝/�Xm,˛ . Then the following statements hold.

(i) Let .�, g/ 2�0, �0Œ�Cm,˛.@˝/. Then equation

�Œ�, g, � , ��D 0

has a unique solution in Xm,˛ , which we denote by . O�Œ�, g�, O�Œ�, g�/ (cf. Lemma 3.4.)(ii) Equation

�Œ0, g0, � , ��D 0

has a unique solution in Xm,˛ , which we denote by . O�Œ0, g0�, O�Œ0, g0�/. Moreover, . O�Œ0, g0�, O�Œ0, g0�/D . Q� , Q�/ (cf. Lemma 3.8.)(iii) �Œ�, �, �, �� is a real analytic map from ��0, �0Œ�Cm,˛.@˝/�Xm,˛ to Cm,˛.@˝/. Moreover, the differential @.� ,/�Œ0, g0, O�Œ0, g0�, O�Œ0, g0��

of� at .0, g0, O�Œ0, g0�, O�Œ0, g0�/with respect to the variables .� , �/ is a linear homeomorphism from Xm,˛ onto Cm,˛.@˝/.(iv) There exist �1 2�0, �0�, an open neighborhood U of g0 in Cm,˛.@˝/, and a real analytic map .�Œ�, ��,�Œ�, ��/ from � � �1, �1Œ�U to

Xm,˛ , such that

.�Œ�, g�,�Œ�, g�/D . O�Œ�, g�, O�Œ�, g�/ 8.�, g/ 2�0, �1Œ�U ,

.�Œ0, g0�,�Œ0, g0�/D . Q� , Q�/ .


34

1

P. MUSOLINO

Proof. Statements (i), (ii) are immediate consequences of Lemmas 3.4, 3.8. We now consider statement (iii). We first introduce somenotation. For each j 2 f1, : : : , ng, we denote by RjŒ�, �� the map from �� 0, �0Œ�L1.@˝/ to Cm,˛.cl˝/, defined by

RjŒ�, f �.t/�

Z@˝.Dxj Rq

n.�.t� s///f .s/d�s 8t 2 cl˝ ,

for all .�, f / 2� � �0, �0Œ�L1.@˝/. By classical potential theory and standard calculus in Banach spaces, we note that the map fromCm,˛.@˝/� Cm,˛.@˝/0 �R to Cm,˛.@˝/, which takes .g, � , �/ to the function

�1

2�.t/�

Z@˝.DSn.t � s//�˝.s/�.s/d�sC � � g.t/

of the variable t 2 @˝ , is linear and continuous, and thus real analytic (cf., e.g., Miranda [35], Dalla Riva, and Lanza [36, Theorem 3.1],Lanza and Rossi [28, Theorem 3.1].) Then, to prove the real analyticity of �Œ�, �, �, �� in � � �0, �0Œ�Cm,˛.@˝/ � Xm,˛ , it clearly suffices toshow that RjŒ�, �� is real analytic in � � �0, �0Œ�L1.@˝/ for each j 2 f1, : : : , ng. Indeed, if RjŒ�, �� is real analytic from � � �0, �0Œ�L1.@˝/ toCm,˛.cl˝/ for all j 2 f1, : : : , ng, then by the continuity of the linear map from Cm,˛.@˝/0 to L1.@˝/ that takes � to .�˝/j� , and by thecontinuity of the trace operator from Cm,˛.cl˝/ to Cm,˛.@˝/, we can deduce the analyticity of�Œ�, �, �, �� in �� 0, �0Œ�Cm,˛.@˝/�Xm,˛ .Now, let id@˝ and idcl˝ denote the identity on @˝ and on cl˝ , respectively. Then we note that the map from ��0, �0Œ to Cm,˛.cl˝ ,Rn/

which takes � to �idcl˝ , and the map from �� 0, �0Œ to Cm,˛.@˝ ,Rn/ that takes � to �id@˝ are real analytic. Moreover,

�cl˝ � �@˝ � .Rn n qZn/[ f0g 8� 2�� 0, �0Œ .

Then by the real analyticity of Dxj Rqn in .Rn n qZn/ [ f0g and by Proposition A.2 (ii) of the Appendix, RjŒ�, �� is real analytic in

� � �0, �0Œ�L1.@˝/, for each j 2 f1, : : : , ng. Hence, �Œ�, �, �, �� is real analytic in � � �0, �0Œ�Cm,˛.@˝/ � Xm,˛ . By standard calculus inBanach spaces, the differential @.� ,/�Œ0, g0, O�Œ0, g0�, O�Œ0, g0�� of � at .0, g0, O�Œ0, g0�, O�Œ0, g0�/ with respect to .� , �/ is delivered by thefollowing formula:

@.� ,/�Œ0, g0, O�Œ0, g0�, O�Œ0, g0��. , �/.t/D�1

2 .t/�

Z@˝.DSn.t� s//�˝.s/ .s/d�sC � 8t 2 @˝ ,

for all . , �/ 2 Xm,˛ . Accordingly, by Proposition A.5 of the Appendix, @.� ,/�Œ0, g0, O�Œ0, g0�, O�Œ0, g0�� is a linear homeomorphism fromXm,˛ onto Cm,˛.@˝/, and the proof of (iii) is complete. Finally, statement (iv) is an immediate consequence of statement (iii) and of theImplicit Function Theorem for real analytic maps in Banach spaces (cf., e.g., Prodi and Ambrosetti [24, Theorem 11.6], Deimling [29, The-orem 15.3].) �

Remark 3.15Let the assumptions of Proposition 3.14 hold. Let �1, U , .�Œ�, ��,�Œ�, ��/ be as in Proposition 3.14 (iv). Then, by the rule of change ofvariables in integrals, by Propositions 2.10, 3.14, and by Lemma 3.4, we have

uŒ�, g�.x/D��n�1Z@˝.DSq

n.x � p� �s//�˝.s/�Œ�, g�.s/d�sC�Œ�, g� 8x 2 SŒ˝� �� ,

for all .�, g/ 2�0, �1Œ�U .

4. A functional analytic representation Theorem for the solution and its energy integral

The following statement shows that suitable restrictions of uŒ�, g�.�/ can be continued real analytically for negative values of �.

Theorem 4.1Let ˛ 2�0, 1Œ. Let m 2 N n f0g. Let assumptions (3.1)–(3.3) hold. Let Qu be as in Lemma 3.8. Let �1, U , �Œ�, �� be as in Proposition 3.14 (iv).Then the following statements hold.

(i) Let V be a bounded open subset of Rn such that clV � Rn n .pC qZn/. Let r 2 N . Then there exist �2 2�0, �1� and a real analyticmap U from �� 2, �2Œ�U to Cr.clV/ such that the following statements hold.(j) clV � SŒ˝� �� for all � 2�� 2, �2Œ.

(jj)

uŒ�, g�.x/D �n�1UŒ�, g�.x/C�Œ�, g� 8x 2 clV , (4.2)

for all .�, g/ 2�0, �2Œ�U . Moreover,

UŒ0, g0�.x/D DSqn.x � p/

Z@˝

�˝.s/g0.s/d�s � DSqn.x � p/

Z@˝

s@Qu

@�˝.s/d�s 8x 2 clV . (4.3)3

42


P. MUSOLINO

(ii) LeteV be a bounded open subset of Rn n cl˝ . Then there exist Q�2 2�0, �1� and a real analytic mapeU from �� Q�2, Q�2Œ�U to Cm,˛.cleV/such that the following statements hold.

(j’) pC �cleV � Q n˝� for all � 2�� Q�2, Q�2Œnf0g.(jj’)

uŒ�, g�.pC �t/DeUŒ�, g�.t/C�Œ�, g� 8t 2 cleV , (4.4)

for all .�, g/ 2�0, Q�2Œ�U . Moreover,

eUŒ0, g0�.t/D Qu.t/ 8t 2 cleV .

Proof. Let Sqn, Rq

n be as in Theorem 2.1. Let .�Œ�, ��,�Œ�, ��/ be as in Proposition 3.14 (iv). We start by proving (i). By taking �2 2�0, �1�

small enough, we can clearly assume that (j) holds. Consider now (jj). If .�, g/ 2�0, �2Œ�U , then by Remark 3.15 we have

uŒ�, g�.x/D��n�1Z@˝.DSq

n.x � p� �s//�˝.s/�Œ�, g�.s/d�sC�Œ�, g� 8x 2 clV .

Thus it is natural to set

UŒ�, g�.x/��

Z@˝.DSq

n.x � p� �s//�˝.s/�Œ�, g�.s/d�s 8x 2 clV ,

for all .�, g/ 2�� 2, �2Œ�U . Then we note that

clV � p� �@˝ �Rn n qZn 8� 2�� 2, �2Œ .

As a consequence, by the real analyticity of Sqn in RnnqZn, and by the real analyticity of�Œ�, �� from ��1, �1Œ�U to Cm,˛.@˝/0, and by

Proposition A.2 (i) of the Appendix, we can conclude that U is real analytic from �� 2, �2Œ�U to Cr.clV/. By the definition of U, equality(4.2) holds for all .�, g/ 2�0, �2Œ�U . Next we turn to prove formula (4.3). First we note that

UŒ0, g0�.x/D�DSqn.x � p/

Z@˝

�˝.s/�Œ0, g0�.s/d�s 8x 2 clV .

Proposition 3.14 (iv) implies that�Œ0, g0�D Q� , where Q� is as in Lemma 3.8. Then we set

w.t/��

Z@˝

�DSn.t� s/

��˝.s/ Q�.s/d�s 8t 2Rn .

As is well known, wj˝ admits a continuous extension to cl˝ , which we denote by wC, and wjRnncl˝ admits a continuous extension

to Rn n˝ , which we denote by w�. Moreover, wC 2 Cm,˛.cl˝/ and w� 2 Cm,˛.Rn n˝/ (cf., e.g., Lanza and Rossi [28, Thm. 3.1].) Clearly,w� D Qu. Then we fix j 2 f1, : : : , ng. By classical potential theory, we haveZ

@˝

��˝.s/

�jQ�.s/d�s D

Z@˝

��˝.s/

�jwC.s/d�s �

Z@˝

��˝.s/

�jw�.s/d�s

(cf., e.g., Lanza and Rossi [28, Thm. 3.1].) Then the Green Identity and classical potential theory imply thatZ@˝

��˝.s/

�jwC.s/d�s D

Z@˝

sj@wC

@�˝.s/d�s D

Z@˝

sj@w�

@�˝.s/d�s

(cf., e.g., Lanza and Rossi [28, Thm. 3.1].) As a consequence, because w� D Qu andR@˝

��˝.s/

�j d�s D 0, we haveZ

@˝

��˝.s/

�jQ�.s/d�s D

Z@˝

sj@Qu

@�˝.s/d�s �

Z@˝.�˝.s//j Qu.s/d�s

D

Z@˝

sj@Qu

@�˝.s/d�s �

Z@˝.�˝.s//jg0.s/d�s .

Accordingly equality (4.3) holds and so the proof of (i) is complete. We now consider (ii). Let R > 0 be such that .cleV[cl˝/� Bn.0, R/.By the continuity of the restriction operator from Cm,˛.clBn.0, R/n˝/ to Cm,˛.cleV/, it suffices to prove statement (ii) witheV replaced byBn.0, R/ n cl˝ . By taking Q�2 2�0, �1� small enough, we can assume that

pC �clBn.0, R/� Q 8� 2�� Q�2, Q�2Œ .

If .�, g/ 2�0, Q�2Œ�U , a simple computation on the basis of the Theorem of change of variables in integrals shows that

uŒ�, g�.pC �t/D�

Z@˝.DSn.t� s//�˝.s/�Œ�, g�.s/d�s � �

n�1Z@˝.DRq

n.�.t� s///�˝.s/�Œ�, g�.s/d�sC�Œ�, g�

8t 2 clBn.0, R/ n cl˝ .


34

3

P. MUSOLINO

Now we set

eGRqnŒ�, g�.t/��n�1

Z@˝.DRq

n.�.t� s///�˝.s/�Œ�, g�.s/d�s 8t 2 clBn.0, R/ n˝ ,

for all .�, g/ 2�� Q�2, Q�2Œ�U . Then we note that

�clBn.0, R/� �@˝ � .Rn n qZn/[ f0g 8� 2�� Q�2, Q�2Œ .

Accordingly, by arguing as in the proof of Proposition 3.14, we can conclude that eGRqnŒ�, �� is real analytic from � � Q�2, Q�2Œ�U to

Cm,˛.clBn.0, R/ n ˝/. Moreover, eGRqnŒ0, g0�.�/ D 0 in clBn.0, R/ n ˝ . By classical results of potential theory and by the real analyticity

of�Œ�, ��, there exists a real analytic mapeGSn Œ�, �� from �� Q�2, Q�2Œ�U to Cm,˛.clBn.0, R/ n˝/, such that

eGSn Œ�, g�.t/D�

Z@˝.DSn.t� s//�˝.s/�Œ�, g�.s/d�s 8t 2 clBn.0, R/ n cl˝ ,

for all .�, g/ 2�� Q�2, Q�2Œ�U (cf., e.g., Miranda [35], Dalla Riva and Lanza [36, Theorem 3.1], Lanza and Rossi [28, Theorem 3.1].) In particular,

eGSn Œ0, g0�.t/D Qu.t/ 8t 2 clBn.0, R/ n˝ .

Then we set

eUŒ�, g�.t/�eGSn Œ�, g�.t/CeGRqnŒ�, g�.t/ 8t 2 clBn.0, R/ n˝ ,

for all .�, g/ 2�� Q�2, Q�2Œ�U . As a consequence,eU is a real analytic map from �� Q�2, Q�2Œ�U to Cm,˛.clBn.0, R/n˝/, such that (jj’) holds witheV replaced by Bn.0, R/ n cl˝ . Thus the proof is complete. �

Remark 4.5Here, we observe that Theorem 4.1 (i) concerns what can be called the ‘macroscopic’ behavior of the solution, whereas Theorem 4.1 (ii)describes the ‘microscopic’ behavior. Indeed, in Theorem 4.1 (i), we consider a bounded open subset V such that clV �Rn n .pC qZn/,that is, such that its closure clV does not intersect the set of points in which the holes degenerate when � tends to 0. Then, for � smallenough, clV is ‘far’ from the union of the holes

Sz2Zn.qz C ˝�/, and we prove a real analytic continuation result for the restriction

of the solution to clV . Instead, in Theorem 4.1 (ii), we take a bounded open subseteV of Rn n cl˝ and we consider the behavior of therestriction of the solution to the set pC �cleV . We note that the set pC �cleV gets, in a sense, closer to the hole ˝� as � tends to 0, andthat it degenerates into the set fpg for � D 0. Therefore, in Theorem 4.1 (ii), we characterize the behavior of the solution in proximity ofthe hole˝� in the fundamental cell Q.

We now turn to consider the energy integral of the solution on a perforated cell, and we prove the following.

Theorem 4.6Let ˛ 2�0, 1Œ. Let m 2 N n f0g. Let assumptions (3.1)–(3.3) hold. Let �1, U be as in Proposition 3.14 (iv). Then there exist �3 2�0, �1� and areal analytic map G from �� 3, �3Œ�U to R, such thatZ

Qncl˝�jDxuŒ�, g�.x/j2 dx D �n�2GŒ�, g� , (4.7)

for all .�, g/ 2�0, �3Œ�U . Moreover,

GŒ0, g0�D

ZRnncl˝

jDQu.t/j2 dt , (4.8)

where Qu is as in Lemma 3.8.

Proof. Let .�, g/ 2�0, �1Œ�U . By the Green Formula and by the periodicity of uŒ�, g�.�/, we haveZQncl˝�

jDxuŒ�, g�.x/j2 dx D

Z@Q

DxuŒ�, g�.x/�Q.x/uŒ�, g�.x/d�x �

Z@˝�

DxuŒ�, g�.x/�˝� .x/uŒ�, g�.x/d�x

D��n�1Z@˝

DxuŒ�, g�.pC �t/�˝.t/g.t/d�t D��n�2

Z@˝

D�

uŒ�, g� ı .pC �idn/�.t/�˝.t/g.t/d�t . (4.9)

Let R > 0 be such that cl˝ � Bn.0, R/. By Theorem 4.1 (ii), there exist �3 2�0, �1� and a real analytic mapeGŒ�, �� from � � �3, �3Œ�U toCm,˛.clBn.0, R/ n˝/, such that

pC �cl.Bn.0, R/ n cl˝/� Q n˝� 8� 2�� 3, �3Œnf0g ,

and that

eGŒ�, g�.t/D uŒ�, g� ı .pC �idn/.t/ 8t 2 clBn.0, R/ n˝ 8.�, g/ 2�0, �3Œ�U ,34

4


P. MUSOLINO

and that

eGŒ0, g0�.t/D Qu.t/C Q� 8t 2 clBn.0, R/ n˝ ,

where Qu, Q� are as in Lemma 3.8. By equality (4.9), we haveZQncl˝�

jDxuŒ�, g�.x/j2 dx D��n�2Z@˝

DteGŒ�, g�.t/�˝.t/g.t/d�t ,

for all .�, g/ 2�0, �3Œ�U . Thus it is natural to set

GŒ�, g��

Z@˝

DteGŒ�, g�.t/�˝.t/g.t/d�t ,

for all .�, g/ 2� � �3, �3Œ�U . Then by continuity of the partial derivatives from Cm,˛.clBn.0, R/ n˝/ to Cm�1,˛.clBn.0, R/ n˝/, and bycontinuity of the trace operator on @˝ from Cm�1,˛.clBn.0, R/ n˝/ to Cm�1,˛.@˝/, and by the continuity of the pointwise product inthe Schauder spaces, and by standard calculus in Banach spaces, we conclude that GŒ�, �� is a real analytic map from �� 3, �3Œ�U to Rand that equality (4.7) holds. Finally, we note that

GŒ0, g0�D�

Z@˝

DQu.t/�˝.t/g0.t/d�t .

By classical potential theory and the Divergence Theorem, we haveZ@˝

DQu.t/�˝.t/d�t D 0 . (4.10)

Then, by the decay properties at infinity of Qu and of its radial derivative and by equality (4.10), we have

�

Z@˝

DQu.t/�˝.t/g0.t/d�t D�

Z@˝

DQu.t/�˝.t/�

g0.t/� Q��

d�t D

ZRnncl˝

jDQu.t/j2 dt

(cf., e.g., Folland [34, p. 118].) As a consequence, equality (4.8) follows, and the proof is complete. �

Remark 4.11In Theorem 4.6, we have shown a real analytic continuation result for the energy integral of the solution on the perforated fundamentalcell Q n cl˝� , which degenerates into the set Q n fpg for � D 0. We note that the energy integral

RQncl˝�

jDxuŒ�, g�.x/j2 dx converges to0 as .�, g/ tends to .0, g0/ if n> 3, whereas in general, this is not true if nD 2. Moreover, because the map from ��3, �3Œ to R that takes� to �n�2GŒ�, g0� is real analytic, Theorem 4.6 implies the existence of �#

3 2�0, �3� and of a sequence of real numbers fajg1jD0 such that

ZQncl˝�

jDxuŒ�, g0�.x/j2 dx D

1XjD0

aj�j 8� 2�0, �#

3Œ ,

where the series converges absolutely in �� #3, �#

3Œ. Clearly, analogous considerations for the ‘macroscopic’ and ‘microscopic’ behaviorof the solution can be derived from the results of Theorem 4.1.

APPENDIX A.

In this Appendix, we collect some results exploited in the article.We have the following known consequence of the Maximum Principle.

Proposition A.1Let I be a bounded connected open subset of Rn such that Rn n clI is connected and that clI � Q. Let u 2 C0.clSŒI��/\ C2.SŒI��/ besuch that

u.xC qei/D u.x/ 8x 2 clSŒI��, 8i 2 f1, : : : , ng ,

and that

�u.x/D 0 8x 2 SŒI�� .

Then the following statements hold.

(i) If there exists a point x0 2 SŒI�� such that u.x0/DmaxclSŒI�� u, then u is constant within SŒI��.(ii) If there exists a point x0 2 SŒI�� such that u.x0/DminclSŒI�� u, then u is constant within SŒI��.


34

5

P. MUSOLINO

(iii)

maxclSŒI��

uDmax@I

u , minclSŒI��

uDmin@I

u .

Proof. Clearly, statement (iii) is a straightforward consequence of (i) and (ii). Furthermore, statement (ii) follows from statement (i) byreplacing u with �u. Therefore, it suffices to prove (i). Let u and x0 be as in the hypotheses. By periodicity of u, supx2SŒI�� u.x/ < C1.Then by the Maximum Principle, u must be constant in SŒI�� (cf., e.g., Folland [34, Theorem 2.13, p. 72].) �

We now introduce the following Proposition on nonlinear integral operators (see [37].)

Proposition A.2Let n, s 2N , 16 s < n. Let M be a compact manifold of class C1 imbedded into Rn and of dimension s. Let K be a Banach space. Let Wbe an open subset of Rn �Rn �K. Let G be a real analytic map from W to R. Then the following statements hold.

(i) Let r 2N . Let˝ be a bounded open subset of Rn. Let

F �n. , z/ 2 C0.M,Rn/�KW cl˝ � .M/� fzg �W

o.

Then the map HG from F � L1.M/ to Cr.cl˝/ defined by

HGŒ , z, f �.x/�

ZM

G.x, .y/, z/f .y/d�y 8x 2 cl˝ ,

for all . , z, f / 2 F � L1.M/ is real analytic.(ii) Let m 2N . Let ˛ 2�0, 1�. Let˝ 0 be a bounded connected open subset of Rn of class C1. Let

F# �n. , , z/ 2 Cm,˛.cl˝ 0,Rn/� C0.M,Rn/�KW .cl˝ 0/� .M/� fzg �W

o.

Let H#G be the map from F# � L1.M/ to Cm,˛.cl˝ 0/ defined by

H#GŒ , , z, f �.t/�

ZM

G. .t/, .y/, z/f .y/d�y 8t 2 cl˝ 0 ,

for all . , , z, f / 2 F# � L1.M/. Then H#G is real analytic from F# � L1.M/ to Cm,˛.cl˝ 0/.

Then we have the following result of (periodic) potential theory (see also Lanza [14, p. 283], Kirsch [38].)

Proposition A.3Let ˛ 2�0, 1Œ, m 2N n f0g. Let Sq

n be as in Theorem 2.1. Let I be a bounded connected open subset of Rn of class Cm,˛ such that Rn n clIis connected and that clI � Q. Let Sq

n be as in Theorem 2.1. Then the following statements hold.

(i) The map wqŒ@I, ��j@I is compact from Cm,˛.@I/ to itself.

(ii) Let eMŒ�� be the map from Cm,˛.@I/ to itself, defined by

eMŒ��.t/��1

2�.t/CwqŒ@I,��.t/ 8t 2 @I ,

for all � 2 Cm,˛.@I/. Then eMŒ�� is a linear homeomorphism from Cm,˛.@I/ onto itself. Moreover,neMŒ��W� 2RoDR , (A.4)

where we identify the constant functions with the constants themselves.

Proof. We start by proving (i). Let Rqn be as in Theorem 2.1. We set

wŒ@I,��.t/��

[email protected]� s//�I.s/�.s/d�s 8t 2 @I ,

for all � 2 Cm,˛.@I/. By classical potential theory and by the compactness of the imbedding of Cm,˛.@I/ into Cm,ˇ .@I/ for ˇ 2�0,˛Œ, weconclude that the operator wŒ@I, �� from Cm,˛.@I/ to itself is compact. Indeed, case mD 1 has been proved by Schauder [39, Hilfsatz XI,p. 618], and case m > 1 follows by taking the tangential derivatives of wŒ@I, �� on @I and by arguing by induction on m. We also set

wRqnŒ@I,��.t/��

[email protected]

n.t� s//�I.s/�.s/d�s 8t 2 @I ,

for all � 2 Cm,˛.@I/. Clearly, wqŒ@I,��D wŒ@I,��CwRqnŒ@I,�� on @I, for all � 2 Cm,˛.@I/. For each j 2 f1, : : : , ng, we set

NRqn ,jŒf �.t/��

[email protected] Rq

n.t� s//f .s/d�s 8t 2 clI ,34

6


P. MUSOLINO

for all f 2 L1.@I/. By Proposition A.2 (i), NRqn ,jŒ�� is linear and continuous from L1.@I/ to CmC1.clI/. Moreover, by the compactness of the

imbedding of CmC1.clI/ into Cm,˛.clI/, NRqn ,jŒ�� is compact from L1.@I/ to Cm,˛.clI/ (cf., e.g., Lanza and Rossi [28, Lemma 2.1].) Then by

the continuity of the map from Cm,˛.@I/ to L1.@I/ that takes � to .�I/j�, and by the continuity of the trace operator from Cm,˛.clI/to Cm,˛.@I/, we immediately deduce the compactness of wR

qnŒ@I, �� from Cm,˛.@I/ to Cm,˛.@I/, and, as a consequence, of wqŒ@I, ��j@I .

Hence the proof of (i) is complete. We now turn to the proof of (ii). By the Open Mapping Theorem, it suffices to prove that eMŒ�� is abijection. By (i) and by the Fredholm Theory, it suffices to show that eMŒ�� is injective. So, let � 2 Cm,˛.@I/ be such that

�1

2�CwqŒ@I,��D 0 on @I .

By Theorem 2.3, w�q Œ@I,�� is a solution of the following problem8<:�u.x/D 0 8x 2 SŒI�� ,u.xC qei/D u.x/ 8x 2 clSŒI��, 8i 2 f1, : : : , ng ,u.x/D 0 8x 2 @I .

As a consequence, by Proposition A.1, w�q Œ@I,��D 0 in clSŒI��. In particular,

@

@�Iw�q Œ@I,��D 0 on @I .

Then, by formula (2.5),

@

@�IwCq Œ@I,��D 0 on @I .

Accordingly, by Theorem 2.3, wCq Œ@I,��jclI 2 Cm,˛.clI/ is a solution of the following problem(�u.x/D 0 8x 2 I ,@@I

u.x/D 0 8x 2 @I .

As a consequence, there exists a constant c 2R such that wCq Œ@I,��D c on clSŒI�. By formula (2.4),

�D wCq Œ@I,��w�q Œ@I,��D c on @I .

Therefore, by formula (2.8),

eMŒ��D w�q Œ@I, c�D�cjIj

jQjon @I ,

and so cD 0. Hence, �D 0. Finally, equality (A.4) follows immediately from equality (2.6). Thus the proof is complete. �

Finally, we have the following well known result of classical potential theory.

Proposition A.5Let ˛ 2�0, 1Œ, m 2 N n f0g. Let ˝ be a bounded connected open subset of Rn of class Cm,˛ . LeteNŒ�, �� be the map from Cm,˛.@˝/0 �Rto Cm,˛.@˝/, defined by

eNŒ�, ��.x/��1

2�.x/�

Z@˝.DSn.x � y//�˝.y/�.y/d�y C � 8x 2 @˝ ,

for all .�, �/ 2 Cm,˛.@˝/0 �R. TheneNŒ�, �� is a linear homeomorphism from Cm,˛.@˝/0 �R onto Cm,˛.@˝/.

Proof. Clearly,eN is linear and continuous (cf., e.g., Miranda [35], Dalla Riva and Lanza [36, Theorem 3.1], Lanza and Rossi [28, Theorem3.1].) By the Open Mapping Theorem, it suffices to show that it is a bijection. By well known results of classical potential theory, we have

Cm,˛.@˝/Dn�

1

2�.�/�

Z@˝.DSn.� � y//�˝.y/�.y/d�yW� 2 Cm,˛.@˝/

o˚< �@˝ > ,

where �@˝ denotes the characteristic function of @˝ (cf., e.g., Folland [34, Ch. 3] and Lanza [10, Appendix A].) On the other hand, as iswell known, for each in the set n

�1

2�.�/�

Z@˝.DSn.� � y//�˝.y/�.y/d�yW� 2 Cm,˛.@˝/

o,


34

7

P. MUSOLINO

there exists a unique � in Cm,˛.@˝/ such that� .x/D� 1

2�.x/�R@˝.DSn.x � y//�˝.y/�.y/d�y 8x 2 @˝ ,R

@˝ �d� D 0

(cf., e.g., Folland [34, Ch. 3] and Lanza [10, Appendix A].) As a consequence, for each 2 Cm,˛.@˝/, there exists a unique pair .�, �/ inCm,˛.@˝/0 �R, such that

.x/D�1

2�.x/�

Z@˝.DSn.x � y//�˝.y/�.y/d�y C � 8x 2 @˝ ,

and soeN is bijective. Thus the proof is complete. �

Acknowledgements

This paper generalizes a part of the work performed by the author in his ‘Laurea Specialistica’ Thesis [40] under the guidance of Profes-sor M. Lanza de Cristoforis. The author wishes to thank Professor M. Lanza de Cristoforis for his constant help during the preparation ofthis paper. The results presented here have been announced in [41]. The author acknowledges the support of the research project ‘Unapproccio funzionale analitico per problemi di omogeneizzazione in domini a perforazione periodica’ of the University of Padova, Italy.

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