A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach M. Lanza de Cristoforis & P. Musolino Abstract: We consider a Neumann problem for the Poisson equation in the periodically perforated Euclidean space. Each periodic perforation has a size proportional to a positive parameter . For each positive and small , we denote by v(, ·) a suitably normalized solution. Then we are interested to analyze the behavior of v(, ·) when  is close to the degenerate value  = 0, where the holes collapse to points. In particular we prove that if n ≥ 3, then v(, ·) can be expanded into a convergent series expansion of powers of  and that if n = 2 then v(, ·) can be expanded into a convergent double series expansion of powers of  and  log . Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis. Keywords: Neumann problem, singularly perturbed domain, periodically perforated domain, real analytic continuation in Banach space. MSC 2010: 35J25; 31B10; 45F15; 47H30

1

Introduction

In this paper we consider a Neumann boundary value problem for the Laplace operator in the periodically perforated Euclidean space Rn . We first introduce the domain of our boundary value problem. We fix once for all n ∈ N \ {0, 1} , and (q11 , . . . , qnn ) ∈]0, +∞[n , and we introduce a periodicity cell Q ≡ Πnj=1 ]0, qjj [ . Then we denote by q the diagonal matrix 

q11  0 q≡  ... 0

0 q22 ... 0

... ... ... ...

 0 0   ...  qnn

and by mn (Q) the n dimensional measure of the fundamental cell Q, and by νQ the outward unit normal to ∂Q, where it exists. Clearly, qZn ≡ {qz : z ∈ Zn } is the set of vertices of a periodic subdivision of Rn corresponding to the fundamental cell Q. Then we consider m ∈ N \ {0} and α ∈]0, 1[ and a subset Ω of Rn satisfying the following assumption. Let Ω be a bounded open connected subset of Rn of class C m,α . Let Rn \ clΩ be connected. Let 0 ∈ Ω .

(1.1)

Next we fix p ∈ Q. Then there exists 0 ∈]0, +∞[ such that p + clΩ ⊆ Q

∀ ∈] − 0 , 0 [ ,

where cl denotes the closure. To shorten our notation, we set Ωp, ≡ p + Ω 1

∀ ∈ R .

(1.2)

Then we introduce the periodic domains [ (qz + Ωp, ) , S[Ωp, ] ≡

S[Ωp, ]− ≡ Rn \ clS[Ωp, ] ,

z∈Zn

for all  ∈]−0 , 0 [. Then a function u from clS[Ωp, ] or from clS[Ωp, ]− to R is q-periodic if u(x+qhh eh ) = u(x) for all x in the domain of u and for all h ∈ {1, . . . , n}. Here {e1 ,. . . , en } denotes the canonical basis of Rn . Next we introduce the data of our problem. To do so, we fix ρ ∈]0, +∞[ and we consider the Roumieu 0 function space Cq,ω,ρ (Rn ) of q-periodic real analytic functions from Rn to R (see (2.1)), and we assume that 0 {f }∈]−0 ,0 [ is a real analytic family in Cq,ω,ρ (Rn ) ,

i.e., that the map from ] − 0 , 0 [ to

0 Cq,ω,ρ (Rn )

(1.3)

which takes  to f is real analytic (see (2.1)), and that

{g }∈]−0 ,0 [ is a real analytic family in C m−1,α (∂Ω) ,

(1.4)

where C m−1,α (∂Ω) denotes the classical Schauder space of exponents m − 1 and α. Then we introduce a constant k ∈ R. (1.5) The role of k is that of a normalizing condition for the solution. Then for each  ∈]0, 0 [, we consider the following Neumann problem.  ∆v(x) = f (x) ∀x ∈ S[Ωp, ]− ,    −  v is q − periodic in clS[Ωp, ] ,
x−p ∂v (1.6) ∀x ∈ ∂Ωp, , ∂νΩp, (x)R= g     1  R v(x) dσx = k , dσ ∂Ωp, ∂Ωp,

where νΩp, denotes the outward unit normal to Ωp, , and where the function v from clS[Ωp, ]− to R represents the unknown of the problem. By applying the Divergence Theorem in Q \ clΩp, , we can easily show that problem (1.6) can have a classical solution only if Z Z f dx + n−1 g dσ = 0 . (1.7) ∂Ω

Q\clΩp,

Then by a standard energy argument and by potential theory, problem (1.6) can be shown to have a unique solution v(, ·) in the space Cqm,α (clS[Ωp, ]− ) of q-periodic functions of class C m,α in clS[Ωp, ]− . Our problem models for example the heat conduction in a periodic porous material. We assume that an isotropic and homogeneous material occupies the region S[Ωp, ]− and we suppose that it is subject to a heat source which is represented by a constant multiple of the function f . Then, by Fourier’s law, the temperature distribution satisfies the Poisson equation with datum f . By means of the Neumann condition, we prescribe the normal component of the heat flux on the boundary of the cavities. Next we pose the following two questions. (j) Let x be a fixed point in Rn \ (p + qZn ). What can be said on the function  7→ v(, x) when  is close to 0 and positive? (jj) Let t be a fixed point in Rn \ Ω. What can be said on the function  7→ v(, p + t) when  is close to 0 and positive? In a sense, question (j) concerns the ‘macroscopic’ behavior of v(, ·), whereas question (jj) concerns the ‘microscopic’ behavior of v(, ·). Questions of this type have long been investigated for linear problems on domains with small holes with the methods of asymptotic analysis, which aim at giving complete asymptotic expansions of the solutions in terms of the parameter . It is perhaps difficult to provide a complete list of the contributions. Here, we mention, e.g., Ammari and Kang [1], Bonnaillie-No¨el, Dambrine, Tordeux, and Vial [2], Maz’ya, Movchan, and Nieves [24], Maz’ya, Nazarov, and Plamenewskij [25], Nazarov and Sokolowski [30], Ward and Keller [34]. Moreover, boundary value problems in domains with periodic inclusions have 2

been analyzed, at least for the two dimensional case, with the method of functional equations (cf. e.g., Castro, Pesetskaya, and Rogosin [3], Drygas and Mityushev [10], Kapanadze, Mishuris, and Pesetskaya [15], Mityushev and Adler [27], Mityushev, Pesetskaya, and Rogosin [28].) Now one could resort to asymptotic analysis and try to answer to question (j) by providing an asymptotic expansion of the type v(, x) =

l X

cj,x, j + o(l )

as  → 0+

j=0

for some l ∈ N and for suitable real numbers {cj,x, }lj=0 (possibly depending on .) Here instead, we wish to represent the functions in (j), (jj) in terms of real analytic maps and in terms of possibly singular at  = 0, but known functions of  (such as −1 , log , etc.). More precisely, in this paper, we answer to question (j), by showing that v(, x) = Vx [, δ2,n  log ] for  positive and small, where Vx is a real analytic function defined on a whole neighborhood of (0, 0) in R2 . Such an equality implies that if n ≥ 3, then we can expand v(, x) in a convergent power series of  for  small and positive, and that if instead n = 2, then we can expand v(, x) in a convergent double power series of ,  log  for  small and positive (see Theorem 5.1 (i).) Then we answer to question (jj) by showing that v(, p + t) = Vtr [, δ2,n  log ]

for  positive and small,

where Vtr is a real analytic function defined on a whole neighborhood of (0, 0) in R2 . Then the same remarks above on the expandability into power series hold also for v(, p + t) (see Theorem 5.1 (ii).) We observe that our approach does have certain advantages. Indeed, the expandability into power series implies the validity of a fully justified asymptotic expansion in powers of  log  and  if n = 2 and n ≥ 3, respectively, and such expansions have coefficients which do not depend upon . In view of possible applications, we point out that our approach provides the indication of which are the correct powers if we wish to look for expansions which yield to convergent power series, and that power series converge rapidly. The above project has been carried out in many papers for the analysis of problems for the Laplace operator in a bounded domain with a small hole (cf. e.g., [16, 18, 19]). Later, such an approach has been extended to problems of linearized elastostatics and to the Stokes’ flow (cf. e.g., [6, 7].) As far as problems in periodically perforated domains are concerned, we mention [8, 22, 29]. We note that our choice of the Schauder functions spaces C m,α for the solutions of our problem is classical. 0 (Rn ) of analytic Instead, we have assumed the Poisson data f to be in the the Roumieu function space Cq,ω,ρ functions in order to ensure the analyticity of the functions Vx , Vtr above. We now briefly outline our strategy. We convert our problem into a system of integral equations by exploiting layer potential representations depending on unknown density functions. Then we observe that by changing the unknown functions appropriately, we can obtain a problem which can be analyzed around the degenerate case  = 0 by means of the Implicit Function Theorem, and we represent the unknowns of the system of integral equations in terms of  and  log . Next we exploit the integral representation of the solutions in terms of the unknowns of the integral equations, and we deduce the representation of the solution in terms of real analytic maps of  and  log .

2

Preliminaries and notation

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 + kykY for all (x, y) ∈ X × Y, while we use the Euclidean norm for Rn . For standard definitions of Calculus in normed spaces, we refer to Deimling [9]. The symbol N denotes the set of natural numbers including 0. Let A be a matrix. Let D ⊆ Rn . Then clD denotes the closure of D and ∂D denotes the boundary of D. We also set D− ≡ Rn \ clD . 3

For all R > 0, x ∈ Rn , xj denotes the j-th coordinate of x, |x| denotes the Euclidean modulus of x in Rn , and Bn (x, R) denotes the ball {y ∈ Rn : |x − y| < R}. Let Ω be an open subset of Rn . The space of m times continuously differentiable real-valued functions on Ω is denoted by C m (Ω, R), or more simply by C m (Ω). r m Let r ∈ N \ {0}.  Let f ∈ (C (Ω)) . The s-th component of f is denoted fs , and Df denotes the Jacobian matrix

∂fs ∂xl

s=1,...,r, . l=1,...,n

Let η ≡ (η1 , . . . , ηn ) ∈ Nn , |η| ≡ η1 + · · · + ηn . Then Dη f denotes

∂ |η| f η n ∂x1 1 ...∂xη n

.

The subspace of C m (Ω) of those functions f whose derivatives Dη f of order |η| ≤ m can be extended with continuity to clΩ is denoted C m (clΩ). The subspace of C m (clΩ) whose functions have m-th order derivatives that are H¨ older continuous with exponent α ∈]0, 1] is denoted C m,α (clΩ) (cf. e.g., Gilbarg and Trudinger [13].) The subspace of C m (clΩ) of those functions f such that f|cl(Ω∩Bn (0,R)) ∈ C m,α (cl(Ω ∩ Bn (0, R))) for all R ∈ r m,α ]0, +∞[ is denoted Cloc (clΩ). Let D ⊆ Rr . Then C m,α (clΩ, D) denotes {f ∈ (C m,α (clΩ)) : f (clΩ) ⊆ D}. We say that a bounded open subset Ω of Rn is of class C m or of class C m,α , if clΩ is a manifold with boundary imbedded in Rn of class C m or C m,α , respectively (cf. e.g., Gilbarg and Trudinger [13, §6.2].) We denote by νΩ the outward unit normal to ∂Ω. For standard properties of functions in Schauder spaces and for the definition of the Schauder spaces on ∂Ω, we refer the reader to Gilbarg and Trudinger [13] (see also [23, §2].) We denote by dσ the area element of a manifold imbedded in Rn . We retain the standard notation for the Lebesgue space Lp (M ) of p-summable functions. Also, if X is a vector subspace of L1 (M ), we find convenient to set   Z X0 ≡ f ∈ X : f dσ = 0 . M

We note that throughout the paper ‘analytic’ means always ‘real analytic’. For the definition and properties of analytic operators, we refer to Deimling [9, §15]. We set δi,j = 1 if i = j, δi,j = 0 if i 6= j for all i, j = 1, . . . , n. If Ω is an arbitrary open subset of Rn , k ∈ N, β ∈]0, 1], we set Cbk (clΩ) ≡ {u ∈ C k (clΩ) : Dγ u is bounded ∀γ ∈ Nn such that |γ| ≤ k} , and we endow Cbk (clΩ) with its usual norm X kukCbk (clΩ) ≡ sup |Dγ u(x)| |γ|≤k

x∈clΩ

∀u ∈ Cbk (clΩ) .

Then we set Cbk,β (clΩ) ≡ {u ∈ C k,β (clΩ) : Dγ u is bounded ∀γ ∈ Nn such that |γ| ≤ k} , and we endow Cbk,β (clΩ) with its usual norm kukC k,β (clΩ) ≡

X

b

|γ|≤k

sup |Dγ u(x)| + x∈clΩ

X

|Dγ u : clΩ|β

∀u ∈ Cbk,β (clΩ) ,

|γ|=k

where |Dγ u : clΩ|β denotes the β-H¨ older constant of Dγ u. Next, we turn to introduce the Roumieu classes. For all bounded open subsets Ω of Rn and ρ > 0, we set   ρ|β| β 0 ∞ 0 kD ukC (clΩ) < +∞ , Cω,ρ (clΩ) ≡ u ∈ C (clΩ) : sup β∈Nn |β|! and 0 (clΩ) ≡ sup kukCω,ρ

β∈Nn

ρ|β| kDβ ukC 0 (clΩ) |β|!

0 ∀u ∈ Cω,ρ (clΩ) ,

  0 0 (clΩ) where |β| ≡ β1 +· · ·+βn for all β ≡ (β1 , . . . , βn ) ∈ Nn . As is well known, the Roumieu class Cω,ρ (clΩ), k · kCω,ρ 0 is a Banach space. By definition, a function u belongs to Cω,ρ (clΩ) if and only if it can be expanded into a convergent Taylor series around each point of clΩ and the radius of convergence of the Taylor series can

4

estimated from below by means of ρ, uniformly at all points of clΩ. We resort to Roumieu spaces because Roumieu spaces are natural classes of functions which generate analytic superposition operators in Schauder spaces, as shown by Preciso [32, Prop. 1.1, p. 101] (see Prop. A.6 of the Appendix.) Next we turn to periodic domains. If Ω is an arbitrary subset of Rn such that clΩ ⊆ Q, then we set [ (qz + Ω) = qZn + Ω , S[Ω]− ≡ Rn \ clS[Ω] . S[Ω] ≡ z∈Zn

If Ω is an open subset of Rn such that clΩ ⊆ Q and if k ∈ N, β ∈]0, 1], then we set  Cqk (clS[Ω]) ≡ u ∈ Cbk (clS[Ω]) : u is q − periodic , which we regard as a Banach subspace of Cbk (clS[Ω]), and n o Cqk,β (clS[Ω]) ≡ u ∈ Cbk,β (clS[Ω]) : u is q − periodic , which we regard as a Banach subspace of Cbk,β (clS[Ω]). The spaces Cqk (clS[Ω]− ) and Cqk,β (clS[Ω]− ) can be defined similarly. If ρ ∈]0, +∞[, then we set   ρ|β| β 0 n ∞ n kD ukC 0 (clQ) < +∞ , (2.1) Cq,ω,ρ (R ) ≡ u ∈ Cq (R ) : sup β∈Nn |β|! where Cq∞ (Rn ) denotes the set of q-periodic functions of C ∞ (Rn ), and ρ|β| 0 kDβ ukC 0 (clQ) ∀u ∈ Cq,ω,ρ (Rn ) . β∈Nn |β|!   0 0 The Roumieu class Cq,ω,ρ (Rn ), k · kCq,ω,ρ (Rn ) is a Banach space. As is well known, there exists a q-periodic tempered distribution Sq,n such that 0 kukCq,ω,ρ (Rn ) ≡ sup

X

∆Sq,n =

δqz −

z∈Zn

1 , mn (Q)

where δqz denotes the Dirac measure with mass in qz (cf. e.g., [20, p. 84].) The distribution Sq,n is determined up to an additive constant, and we can take Sq,n (x) = −

X z∈Zn \{0}

−1 1 e2πi(q z)·x , 2 −1 2 mn (Q)4π |q z|

(2.2)

in the sense of distributions in Rn (cf. e.g., Ammari and Kang [1, p. 53], [20, §3].) Moreover, Sq,n is even, and real analytic in Rn \ qZn , and locally integrable in Rn (cf. e.g., [20, §3].) We observe that formula (2.2) provides the existence of a q-periodic analog of the fundamental solution, but that it is not suitable for computations. Hasimoto [14] has introduced approximation techniques for periodic fundamental solutions by exploiting Ewald’s techniques for evaluating Sq,n (cf., [11].) Cichocki and Felderhof [4] have obtained expressions suitable for computations in the form of rapidly convergent series (see also Sangani, Zhang and Prosperetti [33] and Poulton, Botten, McPhedran, and Movchan [31]). Finally, we note that Mityushev and Adler [27] have proved the validity of a constructive formula for Sq,n in case n = 2 via elliptic functions. Let Sn be the function from Rn \ {0} to R defined by  1 ∀x ∈ Rn \ {0}, if n = 2 , sn log |x| Sn (x) ≡ 1 2−n n |x| ∀x ∈ R \ {0}, if n > 2 , (2−n)sn where sn denotes the (n − 1) dimensional measure of ∂Bn . Sn is well-known to be the fundamental solution of the Laplace operator. 5

Then the function Sq,n − Sn is analytic in (Rn \ qZn ) ∪ {0} (cf. e.g., Ammari and Kang [1, Lemma 2.39, p. 54].) We find convenient to set in (Rn \ qZn ) ∪ {0} .

Rq,n ≡ Sq,n − Sn

(2.3)

Obviously, Rq,n is not a q-periodic function. We note that the following elementary equality holds Sq,n (x) = 2−n Sn (x) +

1 (δ2,n log ) + Rq,n (x) , 2π

(2.4)

for all x ∈ Rn \ −1 qZn and  ∈]0, +∞[. If Ω is a bounded open subset of Rn and f ∈ L∞ (Ω), then we set Z Pn [Ω, f ](x) ≡ Sn (x − y)f (y) dy ∀x ∈ Rn . Ω

If we further assume that Ω ⊆ Q, then we set Z Pq,n [Ω, f ](x) ≡ Sq,n (x − y)f (y) dy

∀x ∈ Rn .

Ω n

Let Ω be a bounded open subset of R of class C 1,α for some α ∈]0, 1[. If H is any of the functions Sq,n , Rq,n and clΩ ⊆ Q or if H equals Sn , we set Z v[∂Ω, H, µ](x) ≡ H(x − y)µ(y) dσy ∀x ∈ Rn , ∂Ω Z ∂ w[∂Ω, H, µ](x) ≡ H(x − y)µ(y) dσy ∀x ∈ Rn , ∂ν Ω (y) ∂Ω Z ∂ H(x − y)µ(y) dσy ∀x ∈ ∂Ω , w∗ [∂Ω, H, µ](x) ≡ ∂ν Ω (x) ∂Ω for all µ ∈ L2 (∂Ω). As is well known, if µ ∈ C 0 (∂Ω), then v[∂Ω, Sq,n , µ] and v[∂Ω, Sn , µ] are continuous in Rn , and we set v + [∂Ω, Sq,n , µ] ≡ v[∂Ω, Sq,n , µ]|clS[Ω]

v − [∂Ω, Sq,n , µ] ≡ v[∂Ω, Sq,n , µ]|clS[Ω]−

v + [∂Ω, Sn , µ] ≡ v[∂Ω, Sn , µ]|clΩ

v − [∂Ω, Sn , µ] ≡ v[∂Ω, Sn , µ]|clΩ− .

Also, if µ is continuous, then w[∂Ω, Sq,n , µ]|S[Ω] admits a continuous extension to clS[Ω], which we denote by w+ [∂Ω, Sq,n , µ] and w[∂Ω, Sq,n , µ]|S[Ω]− admits a continuous extension to clS[Ω]− , which we denote by w− [∂Ω, Sq,n , µ] (cf. e.g., [20, §3].) Similarly, w[∂Ω, Sn , µ]|Ω admits a continuous extension to clΩ, which we denote by w+ [∂Ω, Sn , µ] and w[∂Ω, Sn , µ]|Ω− admits a continuous extension to clΩ− , which we denote by w− [∂Ω, Sn , µ] (cf. e.g., Miranda [26], [23, Thm. 3.1].) In the specific case in which H equals Sn , we omit Sn and we simply write v[∂Ω, µ], w[∂Ω, µ], w∗ [∂Ω, µ] instead of v[∂Ω, Sn , µ], w[∂Ω, R Sn , µ], w∗ [∂Ω, R Sn , µ], respectively. Finally, we denote by −A the integral A divided by the measure of A, for all measurable subsets A of Rn or of a manifold imbedded into Rn .

3

An integral equation formulation for problem (1.6)

First, we face the problem of transforming the nonhomogeneous problem (1.6) into a homogeneous one. The standard way to deal with such issue would be to consider the difference between the unknown v of the nonhomogeneous problem (1.6) and the q-periodic Newtonian potential Pq,n [Q, f ]. Unfortunately however,

6

the compatibility condition (1.7) does not necessarily imply that Newtonian potential Pq,n [Q, f ] satisfies the equation

R

1 mn (Q)

Z

∆Pq,n [Q, f ](x) = f (x) −

Q

f dx = 0 and accordingly the q-periodic

f (y) dy , Q

and not the Poisson equation with datum f (cf. Proposition A.1 of the Appendix.) Then we resort to a different avenue in order to convert (1.6) into a homogeneous problem. To do so, we need the following. Lemma 3.1 Let m ∈ N \ {0}, α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let  ∈]0, 0 [. Then the following statements hold. (i) The function a from clS[Ωp, ] to R defined by Z a (x) ≡ Sn (x − qz − y) dy

∀x ∈ qz + Ωp, ,

Ωp,

for each z ∈ Zn belongs to Cqm+1,α (clS[Ωp, ]) and satisfies the equalities ∆a (x) = 1 and

∂ a (x) = − ∂xj

∀x ∈ S[Ωp, ] ,

(3.1)

Z Sn (x − qz − y)(νΩp, )j (y) dσy

∀x ∈ qz + Ωp, ,

(3.2)

∂Ωp,

for all z ∈ Zn . (ii) The function b from Rn to R defined by Z b (x) ≡

Sq,n (x − y) dy

∀x ∈ Rn ,

Ωp,

belongs to Cq1 (Rn ) and ∂ b (x) = − ∂xj

Z Sq,n (x − y)(νΩp, )j (y) dσy

∀x ∈ Rn .

(3.3)

∂Ωp,

Moreover, b|clS[Ωp, ] ∈ Cqm+1,α (clS[Ωp, ]) , ∆b (x) = 1 −

mn (Ωp, ) mn (Q)

∀x ∈ S[Ωp, ] ,

(3.4) (3.5)

and b|clS[Ωp, ]− ∈ Cqm+1,α (clS[Ωp, ]− ) ,

(3.6)

mn (Ωp, ) mn (Q)

(3.7)

∆b (x) = −

∀x ∈ S[Ωp, ]− .

(iii) Let f ∈ Cqm−1,α (Rn ). Then the function Fi [f ] from clS[Ωp, ] to R defined by Z Z 1 i F [f ](x) ≡ Sq,n (x − y)f (y) dy + a (x) f (y) dy mn (Q) Q Q for all x ∈ clS[Ωp, ] belongs to Cqm+1,α (clS[Ωp, ]) and satisfies the equality ∆(Fi [f ])(x) = f (x) 7

∀x ∈ S[Ωp, ] .

(iv) Let f ∈ Cqm−1,α (Rn ). Then the function Fo [f ] from clS[Ωp, ]− to R defined by Z Z 1 Fo [f ](x) ≡ Sq,n (x − y)f (y) dy − b (x) f (y) dy mn (Ωp, ) Q Q for all x ∈ clS[Ωp, ]− belongs to Cqm+1,α (clS[Ωp, ]− ) and satisfies the equality ∆(Fo [f ])(x) = f (x)

∀x ∈ S[Ωp, ]− .

Proof. We first consider statement (i). The function a is the q-periodic extension of the Newtonian potential corresponding to the moment 1, which is well known to be of class C m+1,α (clΩp, ) and to satisfy equation (3.1) in Ωp, and equation (3.2) on Ωp, for z = 0 (cf. e.g., Gilbarg and Trudinger [13, Lem. 4.1, p. 54, p. 67], [17, p. 855], Miranda [26, 5.II, p. 330].) Hence, a ∈ Cqm+1,α (clS[Ωp, ]) and equations (3.1) and (3.2) hold. Next we consider statement (ii). We have Z Rq,n (x − y) dy ∀x ∈ Rn . (3.8) b (x) = Pn [Ωp, , 1](x) + Ωp,

Since Rq,n is real analytic in (Rn \ qZn ) ∪ {0}, standard results for the Newtonian potential and classical theorems of differentiation under the integral sign imply that b is of class C 1 in an open neighborhood of clQ (cf. e.g., Gilbarg and Trudinger [13, Lem. 4.1, p. 54].) Then by the periodicity of b we deduce that b ∈ Cq1 (Rn ). Moreover, by exploiting the equality in (3.8) in a neighborhood of clQ, by standard results for the Newtonian potential, by classical theorems of differentiation under the integral sign, and by the periodicity of b , we deduce the validity of (3.5), (3.7). By exploiting the equality in (3.8) for x ∈ clQ \ ∂Ωp, , by the Divergence Theorem, by standard results for the first order derivatives of the Newtonian potential ∂ b , we deduce the validity of (3.3). (cf. e.g., [17, p. 855]), and by the continuity and the periodicity of ∂x j m−1,α Since (νΩp, )j ∈ C (∂Ωp, ) for all j ∈ {1, . . . , n}, equality (3.3) and classical properties of the q-periodic ∂ ∂ b|clS[Ωp, ] ∈ Cqm,α (clS[Ωp, ]) and that ∂x b|clS[Ωp, ]− ∈ Cqm,α (clS[Ωp, ]− ) single layer potential imply that ∂x j j (cf. e.g., [20, Thm. 3.7].) Hence, statements (3.4), (3.6) follow. The statements (iii) and (iv) are immediate consequences of (i), (ii), and of Proposition A.1 of the Appendix. 2 We are now ready to convert our problem (1.6) into a homogeneous problem, and we do so by means of the following, which is an immediate consequence of the previous lemma. Theorem 3.2 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let  ∈]0, 0 [. Let (1.3), (1.4), (1.5) hold. Then the function v ∈ Cqm,α (clS[Ωp, ]− ) satisfies problem (1.6) if and only if the function u ∈ Cqm,α (clS[Ωp, ]− ) defined by u(x) ≡ v(x) − Fo [f ](x)

∀x ∈ clS[Ωp, ]− ,

satisfies the following Neumann problem for the Laplace equation  ∀x ∈ S[Ωp, ]− ,   ∆u(x) = 0   is q − periodic in
clS[Ωp, ]− ,   u ∂u  x−p − ∂νΩ∂ (Pq,n [Q, f ](x)) ∂νΩp, (x) = g  p, R 1 ∂  + mn (Ω (P [Ω , 1](x)) f (y) dy ∀x ∈ ∂Ωp, ,  q,n p,  ) ∂ν QR  p, Ωp,   R R f (y) dy R  Q   − − u dσ = k − −∂Ωp, Pq,n [Q, f ] dσ + m P [Ω , 1] dσ . ∂Ωp, ∂Ωp, q,n p, n (Ωp, )

(3.9)

Then we have the following. Theorem 3.3 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let  ∈]0, 0 [. Let (1.3), (1.4), (1.5), (1.7) hold. Then the map u[, ·, ·] from the set of pairs

8

(θ, c) ∈ C m−1,α (∂Ω)0 × R that solve the following integral equations Z 1 n−1 θ(t) + w∗ [∂Ω, θ](t) +  νΩ (t) · DRq,n ((t − s))θ(s) dσs 2 ∂Ω = g (t) − νΩ (t) · DPq,n [Q, f ](p + t) Z 1 − Sn (t − s)νΩ (t) · νΩ (s) dσs mn (Ω) ∂Ω  Z +n−2 Rq,n ((t − s))νΩ (t) · νΩ (s) dσs Z ∂Ω  Z × f (p + s)ds − g dσ ∀t ∈ ∂Ω , Ω ∂Ω Z Z − Sn (t − s)θ(s) dσs dσt  ∂Ω ∂Ω Z Z Z n−1 + − Rq,n ((t − s))θ(s) dσs dσt + c = k − − Pq,n [Q, f ](p + t) dσt ∂Ω ∂Ω ∂Ω   R R f (p + s) ds − ∂Ω g dσ Z  Ω  δ2,n  log  − Pn [Ω, 1](t) + mn (Ω) + mn (Ω) 2π ∂Ω  Z +n−1 Rq,n ((t − s)) ds dσt

(3.10)

(3.11)

Ω

to the set of u ∈ Cqm,α (clS[Ωp, ]− ) which solve problem (3.9), which takes (θ, c) to the function u[, θ, c] defined by   ·−p ](x) + c ∀x ∈ clS[Ωp, ]− , u[, θ, c](x) ≡ v − [∂Ωp, , Sq,n , θ  is a bijection. Proof. Assume that u ∈ Cqm,α (clS[Ωp, ]− ) solves problem (3.9). Then by Lemma A.5 (i) of the Appendix there exists a unique pair (θ, c) ∈ C m−1,α (∂Ω)0 × R such that   ·−p − ](x) + c ∀x ∈ clS[Ωp, ]− . u(x) = v [∂Ωp, , Sq,n , θ  By the third equation of (3.9), we must necessarily have   Z   1 x−p ∂ y−p θ + Sq,n (x − y)θ dσy 2   ∂Ωp, ∂νΩp, (x)   x−p − νΩp, (x) · DPq,n [Q, f ](x) = g  R f dy Q  νΩ (x) · DPq,n [Ωp, , 1](x) ∀x ∈ ∂Ωp, . + mn (Ωp, ) p, Then by setting t =

x−p  ,

1 θ(t) + 2

Z ∂Ω

s= ∂

y−p  ,

we obtain

Sn (t − s)θ(s) dσs + n−1

Z

νΩ (t) · DRq,n ((t − s))θ(s) dσs ∂νΩ (t) ∂Ω = g (t) − νΩ (t) · DPq,n [Q, f ](p + t) R f (y) dy Z Q  − Sq,n ((t − s))νΩ (t) · νΩ (s) dσs ∀t ∈ ∂Ω . mn (Ω) ∂Ω

9

Now by assumption (1.7), we have Z Z f dy = Q

Z f dy +

Ωp,

Z

f dy

f (p + s) dsn − n−1

=

(3.12)

Q\Ωp,

Ω

Z g dσ . ∂Ω

Moreover, Z νΩ (t) · νΩ (s) dσs = 0 ,

(3.13)

∂Ω

and thus we rewrite the third equation of (3.9) in the following form Z ∂ 1 θ(t) + Sn (t − s)θ(s) dσs 2 ∂Ω ∂νΩ (t) Z +n−1 νΩ (t) · DRq,n ((t − s))θ(s) dσs ∂Ω

= g (t) − νΩ (t) · DPq,n [Q, f ](p + t) Z n−2 Sn (t − s)νΩ (t) · νΩ (s) dσs 2−n − mn (Ω) ∂Ω  Z + Rq,n ((t − s))νΩ (t) · νΩ (s) dσs ∂Ω  Z Z × f (p + s) ds − g dσ ∀t ∈ ∂Ω , Ω

∂Ω

which implies the validity of (3.10). By the fourth equation of (3.9), we must necessarily have   Z ·−p 1 v[∂Ωp, , Sq,n , θ ] dσ + c mn−1 (∂Ωp, ) ∂Ωp,  Z =k−− Pq,n [Q, f ](x) dσx ∂Ωp,

R

 Z f dy Z Sq,n (x − y) dy dσx . + − mn (Ωp, ) ∂Ωp, Ωp, Q

Then by setting t =

x−p  ,

s=

y−p  ,

we obtain Z Z

1 mn−1 (∂Ω)

Sq,n ((t − s))θ(s) dσs n−1 dσt + c Z 1 =k− Pq,n [Q, f ](p + t) dσt mn−1 (∂Ω) ∂Ω R Z  f dy Z Q  + − Sq,n ((t − s)) ds dσt . mn (Ω) ∂Ω Ω

Since

R ∂Ω

∂Ω

∂Ω

θ dσ = 0, we can write such an equation in the following form Z Z Z Z − Sn (t − s)θ(s) dσs  dσt + − Rq,n ((t − s))n−1 θ(s) dσs dσt + c ∂Ω ∂Ω ∂Ω ∂Ω Z = k − − Pq,n [Q, f ](p + t) dσt R ∂Ω Z  Z f dy Q  + − Sn (t − s) ds2−n mn (Ω) ∂Ω Ω  Z Z δ2,n log  + ds + Rq,n ((t − s)) ds dσt . 2π Ω Ω

By assumption (1.7), such an equation can be rewritten as equation (3.11).

10

2

4

Analysis of the system of integral equations (3.10), (3.11)

By Theorem 3.3, we are reduced to analyze system (3.10), (3.11) and we now show that for  ∈]0, 0 [, system (3.10), (3.11) has one and only one solution in the unknown (θ, c). Theorem 4.1 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let  ∈]0, 0 [. Let (1.3), (1.4), (1.5), (1.7) hold. Then the system of equations (3.10), (3.11) has one and only one solution (θ[], c[]) ∈ C m−1,α (∂Ω)0 × R. Proof. We first note that the assumptions (1.3), (1.4) and classical regularity properties of layer potentials imply that the right hand side of equation (3.10) belongs to C m−1,α (∂Ω). By Lemma A.7 (i) of the Appendix, also the third summand in the left hand side of (3.10) belongs to C m−1,α (∂Ω). Since Ω− is connected, Proposition A.4 (i) of the Appendix and the rule of change of variable in integrals imply that equation (3.10) R has one and only one solution θ[] ∈ C m−1,α (∂Ω). In order to prove that ∂Ω θ[] dσ = 0, we first prove that the integral on ∂Ω of the right hand side of (3.10) vanishes. To do so, we note that Z n−1 J ≡  g (t) − νΩ (t) · DPq,n [Q, f ](p + t) (4.1) ∂Ω Z 1 Sn (t − s)νΩ (t) · νΩ (s) dσs − mn (Ω) ∂Ω  Z +n−2 Rq,n ((t − s)))νΩ (t) · νΩ (s) dσs  Z ∂Ω  Z × f (p + s) ds − g dσ dσt Ω ∂Ω R   Z f dx ∂ Q  −Pq,n [Q, f ] + Pq,n [Ωp, , 1] dσ = mn (Ωp, ) ∂Ωp, ∂νΩp,   Z x−p + g dσx ,  ∂Ωp, (see (3.10), (3.12).) By Lemma 3.1 (iv), the Laplace operator in Q \ clΩp, applied to the term in braces in the right hand side of (4.1) equals −f , and accordingly the Divergence Theorem and assumption (1.7) imply that   Z Z x−p dσx = 0 . (4.2) J = f dx + g  Q\Ωp, ∂Ωp, R Next we prove that ∂Ω θ[] dσ = 0. The the known identity Z 1 mn (Ωp, ) ∂ Sq,n (x − y) dσy = − ∀x ∈ ∂S[Ωp, ] , 2 mn (Q) ∂Ωp, ∂νΩp, (y) (cf. e.g., [22, Lem. A.1]), and equation (3.10) imply that Z 1 1−n 0 = J  = θ[](t) + w∗ [∂Ω, θ[]](t) 2 ∂Ω Z +n−1 νΩ (t) · DRq,n ((t − s))θ[](s) dσs dσt ∂Ω    Z 1 x−p = 1−n θ[]  ∂Ωp, 2    Z ∂ y−p + Sq,n (x − y)θ[] dσy dσx  ∂Ωp, ∂νΩp, (x)     Z x−p mn (Ωp, ) 1−n = θ[] dσx 1 −  mn (Q) ∂Ωp,   Z mn (Ωp, ) = θ[] dσ 1 − , mn (Q) ∂Ω 11

and thus θ[] ∈ C m−1,α (∂Ω)0 . On the other hand equation (3.11) determines uniquely c[].

2

Then we are ready to introduce the following. Definition 4.2 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let  ∈]0, 0 [. Let (1.3), (1.4), (1.5), (1.7) hold. Let (θ[], c[]) ∈ C m−1,α (∂Ω)0 × R be the only solution of the system of equations (3.10), (3.11). Then we set Z n−1 v(, x) ≡  Sq,n (x − p − s)θ[](s) dσs + c[] ∂Ω R Z f dy Z Q  + Sq,n (x − p − s) ds , Sq,n (x − y)f (y) dy − mn (Ω) Ω Q for all x ∈ clS[Ωp, ]− . By Theorems 3.2, 3.3, 4.1, the function v(, ·) is the only solution of problem (1.6) for each fixed  ∈]0, 0 [. Next, we note that if (θ, c) ∈ C m−1,α (∂Ω)0 × R and if we let  tend to 0 in equations (3.10), (3.11), we obtain a system which we address to as the ‘limiting system’, and which has the following form 1 θ(t) + w∗ [∂Ω, θ](t) = g0 (t) − νΩ (t) · DPq,n [Q, f0 ](p) 2 R Z g dσ ∂Ω 0 + Sn (t − s)νΩ (t) · νΩ (s) dσs ∀t ∈ ∂Ω , mn (Ω) ∂Ω c = k − Pq,n [Q, f0 ](p) ,

(4.3)

(4.4)

(cf. (3.13).) Then we have the following theorem, which shows the unique solvability of system (4.3), (4.4), and its link with a boundary value problem which we shall address to as the ‘limiting boundary value problem’. Theorem 4.3 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let (1.5) hold. Let 0 (Rn ), g0 ∈ C m−1,α (∂Ω). Then the following statements hold. f0 ∈ Cq,ω,ρ ˜ c˜) in the space C m−1,α (∂Ω)0 × R. (i) The limiting system (4.3), (4.4) has one and only one solution (θ, (ii) The ‘limiting boundary value problem’  ∆u = 0     ∂u (t) = g0 (t) − νΩ (t) · DPq,n [Q, f0 ](p) ∂νΩ R g0 dσ R  + ∂Ω S (t − s)νΩ (t) · νΩ (s) dσs  mn (Ω) ∂Ω n   limt→∞ u(t) = 0 ,

in Rn \ clΩ , ∀t ∈ ∂Ω ,

(4.5)

m,α has one and only one solution u ˜ ∈ Cloc (Rn \ Ω). Moreover,

˜ u ˜(t) = v[∂Ω, θ](t)

∀t ∈ Rn \ Ω .

(4.6)

Proof. We first consider statement (i). Since νΩ ∈ C m−1,α (∂Ω, Rn ), standard regularity properties of single R layer potentials imply that ∂Ω Sn (t − s)νΩ (t) · νΩ (s) dσs defines a function of class C m−1,α in the variable t ∈ ∂Ω (cf. e.g., Miranda [26], [23, Thm. 3.1].) Hence, the right hand side of (4.3) belongs to C m−1,α (∂Ω). Since Ω− is connected, classical results of potential theory imply that equation (4.3) has one and only one solution θ˜ ∈ C m−1,α (∂Ω) (cf. e.g., Folland [12, Prop. 3.37], [18, Thm. 5.1 (ii)].) Next we note that the

12

integral of the right hand side of (4.3) equals zero. Indeed, Z  g0 (t) − νΩ (t) · DPq,n [Q, f0 ](p) ∂Ω R  Z g0 dσ Sn (t − s)νΩ (t) · νΩ (s) dσs dσt + ∂Ω mn (Ω) ∂Ω R Z Z g0 dσ = νΩ (t) · DPn [Ω, 1](t) dσt g0 dσ − ∂Ω mn (Ω) ∂Ω ∂Ω R Z Z g dσ ∂Ω 0 = g0 dσ − ∆Pn [Ω, 1] dx = 0 . mn (Ω) Ω ∂Ω Then equation (4.3) and the known identity Z Z ˜ dσ = w∗ [∂Ω, θ] ∂Ω

˜ θw[∂Ω, 1] dσ =

∂Ω

Z

1 θ˜ dσ ∂Ω 2

(4.7)

R imply that ∂Ω θ˜ dσ = 0. Finally equation (4.4) has a unique solution c˜ ∈ R. We now turn to the proof of (ii). We first note that the limiting boundary value problem (4.5) has at most one classical solution (cf. e.g., Folland [12, Prop. 3.4].) Then by classical results of potential theory, m,α we deduce that the function delivered by the right hand side of (4.6) belongs to Cloc (Rn \ Ω) and solves problem (4.5) (cf. e.g., [23, Thm. 3.1].) 2 We are now ready to analyze equations (3.10), (3.11) around the degenerate case in which  = 0. Thus we introduce the following. Theorem 4.4 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let (1.3), (1.4), (1.5) hold. Let (1.7) hold for all  ∈]0, 0 [. Let Λ ≡ (Λj )j=1,2 be the map from ] − 0 , 0 [×R × C m−1,α (∂Ω)0 × R to C m−1,α (∂Ω)0 × R defined by 1 Λ1 [, 1 , θ, c](t) ≡ θ(t) + w∗ [∂Ω, θ](t) Z2 n−1 + νΩ (t) · DRq,n ((t − s))θ(s) dσs ∂Ω

−g (t) + νΩ (t) · DPq,n [Q, f ](p + t) Z 1 + Sn (t − s)νΩ (t) · νΩ (s) dσs mn (Ω) ∂Ω  Z +n−2 Rq,n ((t − s))νΩ (t) · νΩ (s) dσs Z ∂Ω  Z × f (p + s)ds − g dσ ∀t ∈ ∂Ω , Ω

∂Ω

Λ2 [, 1 , θ, c](t) ≡ Z Z Z Z n−1 − Sn (t − s)θ(s) dσs dσt  +  − Rq,n ((t − s)) dσs dσt + c ∂Ω ∂Ω ∂Ω ∂Ω Z −k + − Pq,n [Q, f ](p + t) dσt  ∂Ω  R R Z  f (p + s) ds − g dσ   Ω ∂Ω − − Pn [Ω, 1](t) mn (Ω) ∂Ω  Z 1 + mn (Ω) + n−1 Rq,n ((t − s)) ds dσt , 2π Ω for all (, 1 , θ, c) ∈] − 0 , 0 [×R × C m−1,α (∂Ω)0 × R. Then the following statements hold. 13

(i) Equation Λ[0, 0, θ, c] = 0 is equivalent to the limiting system (4.3), (4.4) and has one and only one ˜ c˜) ∈ C m−1,α (∂Ω)0 × R (cf. Theorem 4.3.) solution (θ, (ii) If  ∈]0, 0 [, then the equation Λ[, δ2,n  log , θ, c] = 0 is equivalent to the system (3.10), (3.11) in the unknown (θ, c), and has one and only one solution (θ[], c[]) ∈ C m−1,α (∂Ω)0 × R. (iii) Λ is real analytic and the image of Λ is contained in C m−1,α (∂Ω)0 × R. ˜ c˜] of Λ at (0, 0, θ, ˜ c˜) with respect to the variable (θ, c) is a homeomorphism (iv) The differential ∂(θ,c) Λ[0, 0, θ, m−1,α from C (∂Ω)0 × R onto itself. ˜ c˜) ∈ C m−1,α (∂Ω)0 × R and a (v) There exist (0 , ] ) ∈]0, 0 [×]0, +∞[ and an open neighborhood U of (θ, 0 0 ] ] real analytic map (Θ[·, ·], C[·, ·]) from ] −  ,  [×] −  ,  [ to U such that δ2,n  log  ∈] − ] , ] [ for all  ∈]0, 0 [, and that the set of zeros of the map Λ in ] − 0 , 0 [×] − ] , ] [×U coincides with the graph of (Θ[·, ·], C[·, ·]). In particular, ˜ c˜) , (Θ[0, 0], C[0, 0]) = (θ, (Θ[, δ2,n  log ], C[, δ2,n  log ]) = (θ[], c[])

∀ ∈]0, 0 [ .

Proof. Statements (i), (ii) are an immediate consequence of the definition of Λ and of Theorems 3.3, 4.3. The analyticity of Λ follows by the linearity and continuity of v[∂Ω, ·]|∂Ω from C m−1,α (∂Ω) to C m,α (∂Ω), and by the linearity and continuity of w∗ [∂Ω, ·] from C m−1,α (∂Ω) to itself, and by assumptions (1.3), (1.4), and by Proposition A.6, and by Lemma A.7 of the Appendix. Next we prove that the image of Λ is contained in C m−1,α (∂Ω)0 × R. Since we have already proved that Λ has values in C m−1,α (∂Ω) × R, it suffices to prove that Z Λ1 [, 1 , θ, c] dσ = 0 , (4.8) ∂Ω m−1,α

for all (, 1 , θ, c) ∈]−0 , 0 [×R×C (∂Ω)0 ×R. We first prove that (4.8) holds with the extra assumption that  > 0. By changing the variable in the integrals and by exploiting the jump formula for the normal derivaR ∂v− [∂Ωp, ,Sq,n ,θ(−1 (·−p))] dσ = 0, tive of a single layer potential, and equations (4.1), (4.2), and equality ∂Q ∂νQ − −1 which follows by the q-periodicity of v [∂Ωp, , Sq,n , θ( (· − p))], we obtain Z Z ∂v − [∂Ωp, , Sq,n , θ(−1 (· − p))] n−1 Λ1 [, 1 , θ, c] dσ = dσ − J ∂νΩp, ∂Ω ∂Ωp, Z ∂v − [∂Ωp, , Sq,n , θ(−1 (· − p))] dσ =− ∂νQ ∂Q Z ∂v − [∂Ωp, , Sq,n , θ(−1 (· − p))] + dσ ∂νΩp, ∂Ωp, Z =− ∆v − [∂Ωp, , Sq,n , θ(−1 (· − p))] dx = 0 . (4.9) Q\clΩp,

R By equality (4.9), the real analytic map ∂Ω Λ1 [·, 1 , θ, c] dσ vanishes in the interval ]0, 0 [. Hence, the real R analytic map ∂Ω Λ1 [·, 1 , θ, c] dσ must vanish in the whole of the interval ] − 0 , 0 [. Next we turn to the proof of statement (iv). By standard calculus in Banach space, the differential of Λ ˜ c˜) with respect to the variable (θ, c) is delivered by the following formula. at (0, 0, θ, ˜ c˜](θ, c) = 1 θ + w∗ [∂Ω, θ] , ∂(θ,c) Λ1 [0, 0, θ, 2 ˜ ∂(θ,c) Λ2 [0, 0, θ, c˜](θ, c) = c , ˜ c˜] is a bijection. To do so, we show that for all (θ, c) ∈ C m−1,α (∂Ω)0 × R. We now show that ∂(θ,c) Λ[0, 0, θ, m−1,α if (h, d) ∈ C (∂Ω)0 × R, then the system  1 ∀t ∈ ∂Ω , 2 θ(t) + w∗ [∂Ω, θ](t) = h(t) (4.10) c = d, 14

has a unique solution (θ, c) ∈ C m−1,α (∂Ω)0 × R. Since Ω− is connected, classical results of potential theory m−1,α (∂Ω) (cf. e.g., Folland [12, imply that the first equation of (4.10) has one and only one solution θ ∈ C R Prop. 3.37], [18, Thm. 5.1 (ii)].) Then equality (4.7) for θ implies that ∂Ω θ dσ = 0. Finally, the second ˜ c˜] is a bijection. Then by the equation in (4.10) delivers c = d. Thus we have proved that ∂(θ,c) Λ[0, 0, θ, ˜ Open Mapping Theorem, the operator ∂(θ,c) Λ[0, 0, θ, c˜] is a homeomorphism, and the proof of statement (iv) is complete. Statement (v) is an immediate consequence of statements (i), (ii), (iii) and (iv) and of the Implicit Function Theorem in Banach spaces (cf. e.g., Deimling [9, Thm. 15.3].) 2

5

A functional analytic representation theorem for the solution of problem (1.6)

We are now ready to prove the following. Theorem 5.1 Let m ∈ N \ {0}, α ∈]0, 1[, ρ ∈]0, +∞[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 be as in (1.2). Let (1.3), (1.4), (1.5) hold. Let (1.7) hold for all  ∈]0, 0 [. Let 0 , ] be as in Theorem 4.4. Then the following statements hold. ˜ be an open subset of Rn with nonzero distance from p + qZn . Then there exist ∗ ∈]0, 0 [ such (i) Let Ω ˜ Ω that ˜ ⊆ S[Ωp, ]− Ω ∀ ∈ [−∗Ω˜ , ∗Ω˜ ] , and Ω˜ ∈]0, ∗Ω˜ [ such that clS[Ωp,∗˜ ]− ⊆ S[Ωp, ]− for all  ∈ [−Ω˜ , Ω˜ ], and a real analytic operator Ω VS[Ωp,∗ ]− from ] − Ω˜ , Ω˜ [×] − ] , ] [ to Cqm,α (clS[Ωp,∗˜ ]− ) such that Ω

˜ Ω

v(, x) = VS[Ωp,∗ ]− [, δ2,n  log ](x) ˜ Ω

∀x ∈ clS[Ωp,∗˜ ]− , Ω

for all  ∈]0, Ω˜ [. Moreover, Z V

S[Ωp,∗ ]− ˜ Ω

Sq,n (x − y)f0 (y) dy ,

[0, 0](x) = c˜ +

(5.1)

Q

for all x ∈ clS[Ωp,∗˜ ]− . Ω

˜ be a bounded open subset of Rn \ clΩ. Then there exist  ˜ ∈]0, 0 [ and a real analytic map V r (ii) Let Ω ˜ Ω,r Ω ] ] m,α ˜ such that from ] − Ω,r (clΩ) ˜ , Ω,r ˜ [×] −  ,  [ to C ˜ ⊆ clS[Ωp, ]− p + clΩ

∀ ∈] − Ω,r ˜ , Ω,r ˜ [,

v(, p + t) = VΩ˜r [, δ2,n  log ](t)

˜, ∀t ∈ clΩ

for all  ∈]0, Ω,r ˜ [. Moreover, VΩ˜r [0, 0](t) = c˜ + Pq,n [Q, f0 ](p)

˜. ∀t ∈ clΩ

(5.2)

Proof. We first prove statement (i). Let ∗Ω˜ , Ω˜ be as in Lemma A.8 (i) of the Appendix. By Definition 4.2 of v(, ·) and by Theorem 4.4, we have Z v(, x) = n−1 Sq,n (x − p − s)Θ[, δ2,n  log ](s) dσs ∂Ω Z +C[, δ2,n  log ] + Sq,n (x − y)f (y) dy Q R f (y) dy Z Q  − Sq,n (x − p − s) ds ∀x ∈ clS[Ωp,∗˜ ]− , Ω mn (Ω) Ω 15

for all  ∈]0, Ω˜ [. Then it is natural to set n−1

Z

VS[Ωp,∗ ]− [, 1 ](x) =  Sq,n (x − p − s)Θ[, 1 ](s) dσs ˜ Ω ∂Ω Z +C[, 1 ] + Sq,n (x − y)f (y) dy Q R f (y) dy Z Q  − Sq,n (x − p − s) ds ∀x ∈ clS[Ωp,∗˜ ]− , Ω mn (Ω) Ω

(5.3)

for all (, 1 ) ∈] − Ω˜ , Ω˜ [×] − ] , ] [. Now it suffices to show that the right hand side of (5.3) defines a real analytic map from ] − Ω˜ , Ω˜ [×] − ] , ] [ to Cqm,α (clS[Ωp,∗˜ ]− ). By the analyticity of the function Ω Sq,n (x − p − s) in the variable (x, , s) when x − p − s ∈ Rn \ qZn , and by standard properties of integral operators with real analytic kernels and with no singularity, and by Theorem 4.4 (v), we conclude R that the function from ] − Ω˜ , Ω˜ [×] − ] , ] [ to Cqm,α (clS[Ωp,∗˜ ]− ) which takes a pair (, 1 ) to the function ∂Ω Sq,n (x − Ω p − s)Θ[, 1 ](s) dσs of the variable x ∈ clS[Ωp,∗˜ ]− is real analytic (cf. e.g., [20, Lem. 7.1 (i), 7.3 (i)], [21, Ω Cor. 3.14].) By the analyticity of the function Sq,n (x − p − s) of the variable (x, , s) when x − p − s ∈ Rn \ qZn and by standard properties of integral operators with real analytic kernels and with no singularity, the map R from ] − Ω˜ , Ω˜ [ to Cqm,α (clS[Ωp,∗˜ ]− ) which takes  to the function Ω Sq,n (x − p − s) ds of the variable Ω x ∈ clS[Ωp,∗˜ ]− is real analytic. Ω Proposition A.2 of the Appendix and assumption (1.3) imply that the map from ]−Ω˜ , Ω˜ [ to Cqm,α (clS[Ωp,∗˜ ]− ) Ω which takes  to Pq,n [Q, f ]|clS[Ωp,∗ ]− is analytic. By the analyticity of C and by assumption (1.3), it follows ˜ Ω

that the right hand side of (5.3) defines an analytic map from ] − Ω˜ , Ω˜ [×] − ] , ] [ to Cqm,α (clS[Ωp,∗˜ ]− ). Ω Next we prove (5.1). By assumptions (1.3) and (1.4), the leftR hand side of (1.7) is continuous in  ∈ ] − 0 , 0 [. Then the validity of (1.7) for all  ∈]0, 0 [ implies that Q f0 dy = 0. Then by Theorem 4.4 (v), we have Z Sq,n (x − y)f0 (y) dy VS[Ωp,∗ ]− [0, 0](x) = C[0, 0] + ˜ Ω Q R f (y) dy Z Q 0 − Sq,n (x − p) ds m (Ω) Ω Z n = c˜ + Sq,n (x − y)f0 (y) dy ∀x ∈ clS[Ωp,∗˜ ]− . Ω

Q

˜ ⊆ Bn (0, R). Then we We now consider statement (ii). By assumption, there exists R > 0 such that clΩ ∗ set Ω ≡ Bn (0, R) \ clΩ. Let Ω∗ ,r be as in Lemma A.8 (ii) of the Appendix with 1 = 0 . Then we take Ω,r ≡ Ω∗ ,r . It clearly suffices to show that VΩr∗ exists and then to set VΩ˜r equal to the composition of the ˜ ˜ with V r∗ . By Definition 4.2 of v(, ·) and by equality restriction of C m,α (clΩ∗ ) to C m,α (clΩ) Ω Z Θ[, δ2,n  log ] dσ = 0 ∀ ∈]0, 0 [ , ∂Ω

16

and by equalities (2.4) and (3.12), we have Z n−1 v(, p + t) =  Sq,n ((t − s))Θ[, δ2,n  log ](s) dσs ∂Ω Z +C[, δ2,n  log ] + Sq,n (p + t − y)f (y) dy Q R f dy Z Q  Sq,n ((t − s)) ds − mn (Ω) Ω Z = Sn (t − s)Θ[, δ2,n  log ](s) dσs ∂Ω Z +n−1 Rq,n ((t − s))Θ[, δ2,n  log ](s) dσs ∂Ω

+C[, δ2,n  log ] + Pq,n [Q, f ](p + t)  Z 1 δ2,n log  2−n −  mn (Ω) Sn (t − s) ds + mn (Ω) 2π Ω Z Z Z n n−1 + Rq,n ((t − s)) ds f (p + s) ds −  Ω

Ω

 g dσ ,

∂Ω

for all t ∈ clΩ∗ . Thus it is natural to set VΩr∗ [, 1 ](t) ≡ v − [∂Ω, Θ[, 1 ]](t) Z n−1 + Rq,n ((t − s))Θ[, 1 ](s) dσs

(5.4)

∂Ω

+C[, 1 ] + Pq,n [Q, f ](p + t)   Z 1 1 mn (Ω) + n−1 Rq,n ((t − s)) ds Pn [Ω, 1](t) + − mn (Ω) 2π Ω  Z  Z ×  f (p + s) ds − g dσ ∀t ∈ clΩ∗ , Ω ]

∂Ω ]

for all (, 1 ) ∈] − Ω,r ˜ , Ω,r ˜ [×] −  ,  [. Now it suffices to prove that the right hand side of (5.4) defines ] ] m,α a real analytic map from ] − Ω,r (clΩ∗ ). By classical potential theory (cf. e.g., ˜ , Ω,r ˜ [ ×] −  ,  [ to C Miranda [26], [23, Thm. 3.1]), by standard properties of integral operators with real analytic kernels and with no singularity (cf. [21, Prop. 4.1 (i)]), by Theorem 4.4, by assumptions (1.3), (1.4), by Proposition A.6 and by Lemma A.7 of the Appendix, and by arguing so as to prove the analyticity of VS[Ωp,∗ ]− , we conclude ˜ Ω

] ] m,α that the right hand side of (5.4) defines a real analytic function from ] − Ω,r (clΩ∗ ). ˜ , Ω,r ˜ [×] −  ,  [ to C Moreover, Theorem 4.4 (v) implies the validity of (5.2). 2

6

Conclusions

In the present paper, we have considered a Neumann problem for the Poisson equation in a periodically perforated domain obtained by making a periodic set of holes in the Euclidean space Rn . The equal size of the periodic perforations is determined by a small positive parameter . Such a problem models the heat conduction in a porous material with prescribed heat flux on the boundary of the holes. Our aim is to investigate the asymptotic behavior of a suitable normalized solution v(, ·). By Theorem 5.1, if x ∈ Rn \ (p + qZn ) we can deduce the existence of a real analytic function Vx defined on a neighborhood of (0, 0) in R2 such that v(, x) = Vx [, δ2,n  log ] for  small and positive. In particular, if n ≥ 3, such an equality implies the existence of a sequence of real numbers {cj,x }∞ j=0 such that ∞ X v(, x) = cj,x j for  small and positive, j=0

17

where the series converges absolutely in a convenient neighborhood of 0. Therefore, it is relevant to provide a method to compute the coefficients cj,x . The authors are investigating such issue and results on a fully constructive method for the computation of cj,x via integral equations for problems of this type will appear in forthcoming papers. A natural development of the present paper is the analysis of homogenization problems in an infinite periodically perforated domain, when also the size of the periodicity cell tends to 0. The approach adopted in the present paper has been exploited to analyze linear and nonlinear boundary problems for the Poisson and the Laplace equations in S[Ωp, ]− . The authors plan to extend such techniques to different problems, including the behavior of the eigenvalues of the Laplace operator in a periodically perforated domain and the Stokes’ flow past a periodic array of small obstacles.

A

Appendix

We first introduce two statements concerning periodic volume potentials. Proposition A.1 Let m ∈ N \ {0}, α ∈]0, 1[. Let f ∈ Cqm−1,α (Rn ). Then Pq,n [Q, f ] ∈ Cqm+1,α (Rn ) and Z 1 f (y) dy ∀x ∈ Rn . (A.1) ∆Pq,n [Q, f ](x) = f (x) − mn (Q) Q Proof. Clearly, Pq,n [Q, f ] is q-periodic. We first show that Pq,n [Q, f ] ∈ C 2 (Rn ) and that (A.1) holds. Then, by standard regularity theory for the solutions of the Poisson equation, we deduce that Pq,n [Q, f ] ∈ Cqm+1,α (Rn ) (cf. e.g., Gilbarg and Trudinger [13, Thm. 4.6, p. 60].) So let x0 ∈ Rn and set Ux0 ≡ ˜ x ≡ Qn ] − qjj , qjj [+x0 . Clearly, clUx ⊆ Q ˜ x . By the periodicity Bn (x0 , min{q11 , . . . , qnn }/3) and Q 0 0 0 j=1 2 2 ˜ x , f ] (cf. e.g., Cioranescu and Donato [5, Lem. 2.3, p. 27].) We of Sq,n and f , we have Pq,n [Q, f ] = Pq,n [Q 0 R now show that Pq,n [Q, f ]|Ux0 ∈ C 2 (Ux0 ) and that ∆Pq,n [Q, f ] = f − −Q f dy in Ux0 . In order to do so, we note that if we set Z ˜ x , f ](x) , u1 (x) ≡ Pn [Q u (x) ≡ Rq,n (x − y)f (y) dy ∀x ∈ Ux0 , 2 0 ˜x Q 0

we have Pq,n [Q, f ] = u1 +u2 in Ux0 . By Gilbarg and Trudinger [13, Lemma 4.2, p. 55], we have u1 ∈ C 2 (Ux0 ) ˜ x , then (x − y) ∈ (Rn \ qZn ) ∪ {0}. Since Rq,n and ∆u1 = f in Ux0 . Next we note that if x ∈ Ux0 , y ∈ Q 0 n n is analytic in (R \ qZ ) ∪ {0}, classical theoremsR of differentiation under the integral sign imply that R u2 ∈ C ∞ (Ux0 ) and that ∆u2 = −−Q˜ x f dy = −−Q f dy in Ux0 . Hence, Pq,n [Q, f ]|Ux0 ∈ C 2 (Ux0 ) and 0 R 2 ∆Pq,n [Q, f ] = f − −Q f dy in Ux0 . 0 Proposition A.2 Let ρ > 0. Then there exists ρ0 ∈]0, ρ] such that the linear map from Cq,ω,ρ (Rn ) to 0 n Cq,ω,ρ0 (R ) which takes f to Pq,n [Q, f ] is continuous.

Proof. It clearly suffices to show that for each x0 ∈ Rn , there exist rx0 , ρx0 ∈]0, +∞[, such that Pq,n [Q, ·]|clBn (x0 ,rx0 ) 0 0 is a continuous operator from Cq,ω,ρ (Rn ) to Cq,ω,ρ (clBn (x0 , rx0 )). Then by considering a convenient fix0 nite open covering of clQ, we can deduce the validity of the statement. So we now fix x0 ∈ Rn and set Q q q n jj jj ˜x ≡ Q ˜x0 ≡ min{q11 , . . . , qnn }/3. Since Pq,n [Q, f ] = 0 j=1 ] − 2 , 2 [+x0 , rx0 ≡ min{q11 , . . . , qnn }/4, and r ˜ Pq,n [Qx0 , f ], we have Z Pq,n [Q, f ](x) =Pn [Bn (x0 , r˜x0 ), f ](x) + Sn (x − y)f (y) dy ˜ x \clBn (x0 ,˜ Q rx0 ) 0

Z + ˜x Q 0

Rq,n (x − y)f (y) dy

(A.2)

∀x ∈ clBn (x0 , rx0 ) ,

0 for all f ∈ Cq,ω,ρ (Rn ). By [17, Prop. 2.4], there exists ρ˜x0 ∈]0, ρ] such that the linear map Pn [Bn (x0 , r˜x0 ), ·]|clBn (x0 ,rx0 ) 0 0 is continuous from Cq,ω,ρ (Rn ) to Cω, ρ˜x (clBn (x0 , rx0 )). Then by equality (A.2), by standard properties of 0

18

integral operators with real analytic kernels and with no singularity (cf. e.g., [21, §3]), and by the analyticity of Sn , Rq,n in Rn \ {0} and (Rn \ qZn ) ∪ {0}, respectively, we deduce that there exists ρx0 ∈]0, ρ˜x0 ] such that 0 0 Pq,n [Q, ·]|clBn (x0 ,rx0 ) is continuous from Cq,ω,ρ (Rn ) to Cq,ω,ρ (clBn (x0 , rx0 )). 2 x0 Next we introduce the following known slight variant of a classical result in Potential Theory, whose mn (Ω) on ∂S[Ω] and on classical uniqueness results for the proof is based on the identity w[∂Ω, Sq,n , 1] = 21 − m n (Q) periodic Dirichlet and Neumann problems (see [22, Prop. A4].) Proposition A.3 Let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α such that clΩ ⊆ Q. Then the following statements hold. (i) If Rn \ clΩ is connected, and if µ ∈ L2 (∂Ω), and if either Z 1 ∂ µ(t) + µ(s) (Sq,n (t − s)) dσs = 0 2 ∂ν (t) Ω ∂Ω or

1 µ(t) + 2

Z µ(s) ∂Ω

∂ (Sq,n (t − s)) dσs = 0 ∂νΩ (s)

for a.a. t ∈ ∂Ω ,

for a.a. t ∈ ∂Ω ,

then µ = 0. (ii) If Ω is connected, and if µ ∈ L2 (∂Ω), and if either Z 1 ∂ µ(s) (Sq,n (t − s)) dσs = 0 − µ(t) + 2 ∂ν Ω (t) ∂Ω or

1 − µ(t) + 2

Z µ(s) ∂Ω

∂ (Sq,n (t − s)) dσs = 0 ∂νΩ (s)

for a.a. t ∈ ∂Ω ,

for a.a. t ∈ ∂Ω ,

then µ = 0. Then by exploiting equality (2.3), and by the Fredholm Theory, and by the Schauder regularity theory (see [22, Thm. A.2]), and by Proposition A.3, we immediately deduce the validity of the following. Proposition A.4 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C m,α such that clΩ ⊆ Q. Then the following statements hold. (i) If Rn \ clΩ is connected and if Γ ∈ C m−1,α (∂Ω), then the equation Z ∂ 1 µ(t) + µ(s) (Sq,n (t − s)) dσs = Γ(t) 2 ∂νΩ (t) ∂Ω

for a.a. t ∈ ∂Ω ,

has one and only one solution µ ∈ C m−1,α (∂Ω). (ii) If Rn \ clΩ is connected and if Γ ∈ C m,α (∂Ω), then the equation Z 1 ∂ µ(t) + µ(s) (Sq,n (t − s)) dσs = Γ(t) 2 ∂νΩ (s) ∂Ω

for a.a. t ∈ ∂Ω ,

has one and only one solution µ ∈ C m,α (∂Ω). (iii) If Ω is connected and Γ ∈ C m−1,α (∂Ω), then the equation Z 1 ∂ − µ(t) + µ(s) (Sq,n (t − s)) dσs = Γ(t) 2 ∂ν Ω (t) ∂Ω has one and only one solution µ ∈ C m−1,α (∂Ω).

19

for a.a. t ∈ ∂Ω ,

(iv) If Ω is connected and Γ ∈ C m,α (∂Ω), then the equation Z 1 ∂ − µ(t) + (Sq,n (t − s)) dσs = Γ(t) µ(s) 2 ∂νΩ (s) ∂Ω

for a.a. t ∈ ∂Ω ,

has one and only one solution µ ∈ C m,α (∂Ω). Next we introduce the following representation theorem for periodic harmonic functions. Lemma A.5 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C m,α such that clΩ ⊆ Q. Then the following statements hold. (i) Let Rn \clΩ be connected. Then the map from C m−1,α (∂Ω)0 ×R to the Banach subspace of Cqm,α (clS[Ω]− ) of those functions which are harmonic in S[Ω]− which takes a pair (µ, c) to v − [∂Ω, Sq,n , µ] + c is a linear homeomorphism. (ii) Let Ω be connected. Then the map from C m−1,α (∂Ω)0 × R to the Banach subspace of Cqm,α (clS[Ω]) of those functions which are harmonic in S[Ω] which takes a pair (µ, c) to v + [∂Ω, Sq,n , µ] + c is a linear homeomorphism. Proof. Statement (i) can be proved as in [22, Lem. 3.2]. Statement (ii) follows by Proposition A.4 (iii) and by standard results on the periodic single layer potential and on boundary value problems in periodic domains (cf. e.g., [20, Thm. 3.7 (i), (ii)].) 2 Then we have the following variant of a result of Preciso [32, Prop. 1.1, p. 101]. Proposition A.6 Let n1 , n2 ∈ N \ {0}, ρ ∈]0, +∞[, m ∈ N, α ∈]0, 1]. Let Ω1 be a bounded open subset of Rn1 . Let Ω2 be a bounded open connected subset of Rn2 of class C 1 . Then the composition operator T from 0 Cω,ρ (clΩ1 ) × C m,α (clΩ2 , Ω1 ) to C m,α (clΩ2 ) defined by 0 ∀(u, v) ∈ Cω,ρ (clΩ1 ) × C m,α (clΩ2 , Ω1 ) ,

T [u, v] ≡ u ◦ v is real analytic.

Next we introduce the following technical lemma. Lemma A.7 Let m ∈ N \ {0}, α ∈]0, 1[. Let Ω] be a bounded open subset of Rn of class C 1 . Then the following statements hold. (i) Let Ω be as in (1.1). Let ] ∈]0, +∞[ be such that (clΩ] − clΩ) ⊆ (Rn \ qZn ) ∪ {0}

∀ ∈] − ] , ] [ .

Then the maps from ] − ] , ] [×L1 (∂Ω) to C m,α (clΩ] ) which take (, θ) to the functions R R ((t − s))θ(s) dσs ∀t ∈ clΩ] , ∂Ω q,n R ∂ R ((t − s))θ(s) dσs ∀j ∈ {1, . . . , n} ∀t ∈ clΩ] , ∂Ω xj q,n respectively, and the map from ] − ] , ] [ to C m,α (clΩ] ) which takes  to the function Z Rq,n ((t − s)) ds ∀t ∈ clΩ] ,

(A.3)

(A.4) (A.5)

(A.6)

Ω

are real analytic. 0 (ii) Let ρ ∈]0, +∞[. Let p ∈ Q. Then the maps from R × Cq,ω,ρ (Rn ) to C m,α (clΩ] ) which take (, f ) to the functions

Pq,n [Q, f ](p + t) ∂xj Pq,n [Q, f ](p + t) Z

∀t ∈ clΩ] , ∀j ∈ {1, . . . , n}

(A.7) ]

∀t ∈ clΩ ,

(A.8)

1

∂xj Pq,n [Q, f ](p + tτ ) dτ 0

are real analytic, respectively. 20

∀j ∈ {1, . . . , n} ∀t ∈ clΩ] ,

(A.9)

Proof. We first consider statement (i) and prove the analyticity of the map delivered by (A.4). Let id∂Ω and idclΩ] denote the identity map in ∂Ω and clΩ] , respectively. Then we note that the maps from ] − ] , ] [ ] to C m,α (∂Ω, Rn ) and to C m,α (clΩ , Rn ) which take  to id∂Ω and to idclΩ] are real analytic, respectively. By the real analyticity of Rq,n in (Rn \ qZn ) ∪ {0}, by (A.3), and by a result on integral operators with analytic kernels of [21, Prop. 4.1 (i)], the map in (A.4) is analytic from ] − ] , ] [×L1 (∂Ω) to C m,α (clΩ] ). The proof of the analyticity of the maps in (A.5), (A.6), respectively, follows the lines of the proof of the analyticity of the map delivered by (A.4). Next we consider statement (ii) and we prove the analyticity of the map delivered by (A.7). By Proposition 0 0 n A.2 there exists ρ0 ∈]0, ρ] such that the linear map from Cq,ω,ρ (Rn ) to Cq,ω,ρ 0 (R ) which takes f to Pq,n [Q, f ] 0 is continuous. Then by Proposition A.6, the map from R × Cq,ω,ρ (Rn ) to C m,α (clΩ] ) delivered by (A.7). We now consider the analyticity of the map delivered by (A.8). As it can be readily verified, there exists 0 n 0 n 0 < ρ˜ ≤ ρ0 such that ∂xj is continuous from Cq,ω,ρ 0 (R ) to Cq,ω,ρ ˜(R ). Then, by arguing so as to prove the analyticity of the map in (A.7), we deduce the analyticity of the map delivered by (A.8). We now consider the 0 analyticity of the map delivered by (A.9). By Propositions A.2 and A.6, the map from Rn × R × Cq,ω,ρ (Rn ) to R which takes (t, ξ, f ) to ∂xj Pq,n [Q, f ](p + ξt) is real analytic. Then by a result on integral operators with real analytic kernels and with no singularity (cf. [21, Prop. 4.1 (i)]), we deduce the analyticity of the map delivered by (A.9). 2 Next we introduce the following elementary lemma of [22, Lem. A.5]. Lemma A.8 Let m ∈ N \ {0}, α ∈]0, 1[. Let p ∈ Q. Let Ω be as in (1.1). Let 0 ∈]0, +∞[ be as in (1.2). Let 1 ∈]0, 0 [. ˜ be an open subset of Rn with a nonzero distance from p + qZn . Then there exist ∗ ∈]0, 1 [ such (i) Let Ω ˜ Ω that ˜ ⊆ S[Ωp, ]− Ω ∀ ∈ [−∗Ω˜ , ∗Ω˜ ] , and Ω˜ ∈]0, ∗Ω˜ [ such that

clS[Ωp,∗˜ ]− ⊆ S[Ωp, ]− Ω

∀ ∈ [−Ω˜ , Ω˜ ] .

(ii) Let Ω] be a bounded open subset of Rn such that Ω] ⊆ Rn \ clΩ. Then there exists Ω] ,r ∈]0, 1 [ such that p + clΩ] ⊆ Q , p + Ω] ⊆ S[Ωp, ]− ∀ ∈ [−Ω] ,r , Ω] ,r ] \ {0} .

Acknowledgement This paper represents an extension of a part of the work performed by P. Musolino in his Doctoral Dissertation at the University of Padova under the guidance of M. Lanza de Cristoforis. The authors acknowledge the support of “Progetto di Ateneo: Singular perturbation problems for differential operators – CPDA120171/12” - University of Padova and of Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The research of P. Musolino was partially supported by the scholarship “Royal Society” of the “Accademia Nazionale dei Lincei”.

References [1] H. Ammari and H. Kang. Polarization and moment tensors. Applied Mathematical Sciences, 162. Springer, New York, 2007. [2] V. Bonnaillie-No¨el, M. Dambrine, S. Tordeux, and G. Vial. Interactions between moderately close inclusions for the Laplace equation. Math. Models Methods Appl. Sci., 19, (2009), pp. 1853–1882. [3] L.P. Castro, E. Pesetskaya, and S.V. Rogosin. Effective conductivity of a composite material with nonideal contact conditions. Complex Var. Elliptic Equ., 54, (2009), pp. 1085–1100.

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[4] B. Cichocki and B.U. Felderhof. Periodic fundamental solution of the linear Navier-Stokes equations. Phys. A, 159, (1989), pp. 19–27. [5] D. Cioranescu and P. Donato. An introduction to homogenization. Oxford Lecture Series in Mathematics and its Applications, 17. The Clarendon Press Oxford University Press, New York, 1999. [6] M. Dalla Riva. Stokes flow in a singularly perturbed exterior domain. Complex Var. Elliptic Equ., 58, (2013), pp. 231–257. [7] M. Dalla Riva and M. Lanza de Cristoforis. Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: a functional analytic approach. Complex Anal. Oper. Theory, 5, (2011), pp. 811–833. [8] M. Dalla Riva and P. Musolino. A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite. SIAM J. Appl. Math., 73, (2013), pp. 24–46. [9] K. Deimling. Nonlinear functional analysis. Springer-Verlag, Berlin, 1985. [10] P. Drygas and V. Mityushev. Effective conductivity of unidirectional cylinders with interfacial resistance. Quart. J. Mech. Appl. Math., 62, (2009), pp. 235–262. [11] P.P. Ewald. Die Berechnung optischer und elektrostatischer Gitterpotentiale. Ann. Phys., 369, (1921), pp. 253–287. [12] G.B. Folland. Introduction to partial differential equations. Princeton University Press, Princeton, NJ, second edition, 1995. [13] D. Gilbarg and N.S. Trudinger. Elliptic partial differential equations of second order. Springer Verlag, Berlin, etc., 1983. [14] H. Hasimoto. On the periodic fundamental solutions of the Stokes’ equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech., 5, (1959), pp. 317–328. [15] D. Kapanadze, G. Mishuris, and E. Pesetskaya. Improved algorithm for analytical solution of the heat conduction problem in doubly periodic 2D composite materials. Complex Var. Elliptic Equ., to appear. [16] M. Lanza de Cristoforis. Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces. Comput. Methods Funct. Theory, 2, (2002), pp. 1–27. [17] M. Lanza de Cristoforis. A domain perturbation problem for the Poisson equation. Complex Var. Theory Appl., 50, (2005), pp. 851–867. [18] M. Lanza de Cristoforis. Asymptotic behavior of the solutions of a nonlinear Robin problem for the Laplace operator in a domain with a small hole: a functional analytic approach. Complex Var. Elliptic Equ., 52, (2007), pp. 945–977. [19] M. Lanza de Cristoforis. Simple Neumann eigenvalues for the Laplace operator in a domain with a small hole A functional analytic approach. Rev. Mat. Complut., 25, (2012), pp. 369–412 [20] M. Lanza de Cristoforis and P. Musolino. A perturbation result for periodic layer potentials of general second order differential operators with constant coefficients. Far East J. Math. Sci. (FJMS), 52, (2011), pp. 75–120. [21] M. Lanza de Cristoforis and P. Musolino. A real analyticity result for a nonlinear integral operator. J. Integral Equations Appl., 25, (2013), pp. 21–46. [22] M. Lanza de Cristoforis and P. Musolino. A singularly perturbed nonlinear Robin problem in a periodically perforated domain: a functional analytic approach. Complex Var. Elliptic Equ., 58, (2013), pp. 511–536. [23] M. Lanza de Cristoforis and L. Rossi. Real analytic dependence of simple and double layer potentials upon perturbation of the support and of the density. J. Integral Equations Appl., 16, (2004), pp. 137–174. 22

[24] V. Maz’ya, A. Movchan, and M. Nieves. Green’s kernels and meso-scale approximations in perforated domains. Lecture Notes in Mathematics, 2077. Springer, Berlin, 2013. [25] V. Maz’ya, S. Nazarov, and B. Plamenevskij. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vols. I, II, Operator Theory: Advances and Applications, 111, 112. Birkh¨ auser Verlag, Basel, 2000. [26] C. Miranda. Sulle propriet` a di regolarit` a di certe trasformazioni integrali. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I, 7, (1965), pp. 303–336. [27] V. Mityushev and P.M. Adler. Longitudinal permeability of spatially periodic rectangular arrays of circular cylinders. I. A single cylinder in the unit cell. ZAMM Z. Angew. Math. Mech., 82, (2002), pp. 335– 345. [28] V.V. Mityushev, E. Pesetskaya, and S.V. Rogosin. Analytical Methods for Heat Conduction in Composites and Porous Media, in Cellular and Porous Materials: Thermal Properties Simulation and Predic¨ tion (eds A. Ochsner, G.E. Murch and M.J.S. de Lemos), pp. 121–164, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008. [29] P. Musolino. A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach. Math. Methods Appl. Sci., 35, (2012), pp. 334–349. [30] S.A. Nazarov and J. Sokolowski. Asymptotic analysis of shape functionals. J. Math. Pures Appl. (9), 82, (2003), pp. 125–196. [31] C.G. Poulton, L.C. Botten, R.C. McPhedran, and A.B. Movchan. Source-neutral Green’s functions for periodic problems in electrostatics, and their equivalents in electromagnetism. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455, (1999), pp. 1107–1123. [32] L. Preciso. Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Romieu type spaces, Tr. Inst. Mat. Minsk, 5, (2000), pp. 99–104. [33] A.S. Sangani, D.Z. Zhang, and A. Prosperetti. The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small-amplitude oscillatory motion. Phys. Fluids A. 3, (1991), pp. 2955–2970. [34] M.J. Ward and J.B. Keller. Strong localized perturbations of eigenvalue problems. SIAM J. Appl. Math., 53, (1993), pp. 770–798.

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