ENERGY INTEGRAL OF A NONLINEAR TRACTION PROBLEM IN A SINGULARLY PERTURBED PERIODICALLY PERFORATED DOMAIN Matteo Dalla Riva and Paolo Musolino Abstract In this paper, we consider a nonlinear traction problem for the Lam´e equations in a periodically perforated domain obtained by making in Rn a periodic set of holes, each of them of size proportional to . Under suitable assumptions, we know that there exists a family of solutions {u(, ·)}∈]0,1 [ with a prescribed limiting behavior when  approaches 0. Then we investigate the energy integral of u(, ·) as  tends to 0, and we prove that such integral can be continued real analytically for negative values of .

Keywords: Nonlinear traction problem, Energy integral, Singularly perturbed periodic domain, Linearized elastostatics, Real analytic continuation 2010 Mathematics Subject Classification: 35J65, 31B10, 45F15, 74B05

1

INTRODUCTION

In this paper, we present an approach based on potential theory and on functional analysis in order to analyze a singularly perturbed nonlinear traction problem. We fix once for all a natural number n ∈ N \ {0, 1} , positive real numbers q11 , . . . , qnn ∈]0, +∞[ , and a periodicity cell Q ≡ Πnj=1 ]0, qjj [ . Then we denote by q the n × n diagonal matrix defined by  q11 0 . . .  0 q22 . . .  q≡ ... ... ... 0 0 ...

 0 0  . ...  qnn

Now let m ∈ N\{0} and α ∈]0, 1[. Then we consider a subset Ωh of Rn satisfying the following assumption. Ωh is a bounded open connected subset of Rn of class C m,α which contains the origin 0 and with a connected exterior Rn \ clΩh .

(1.1)

Here cl denotes the closure and the letter ‘h’ stands for ‘hole’. The set Ωh will play the role of the shape of the perforation in the periodicity cell Q. For the definition of functions and sets of the usual Schauder

1

k,α classes C k,α and Cloc , with k ∈ N, we refer for example to Gilbarg and Trudinger [19, §6.2] (see also [16, §2]). Next we fix p ∈ Q and we observe that we can find 0 ∈]0, +∞[ such that

p + clΩh ⊆ Q

∀ ∈] − 0 , 0 [ .

(1.2)

To shorten our notation, we set Ωhp, ≡ p + Ωh and we define the periodically perforated domain S[Ωhp, ]− ≡ Rn \ ∪z∈Zn cl(Ωhp, + qz) for all  ∈] − 0 , 0 [. A function u defined on clS[Ωhp, ]− is said to be q-periodic if ∀x ∈ clS[Ωhp, ]− ,

u(x + qej ) = u(x)

for all j ∈ {1, . . . , n}. Here {e1 ,. . . , en } denotes the canonical basis of Rn . We now introduce a nonlinear traction boundary value problem in S[Ωhp, ]− . To do so, we denote by T the function from ]1 − (2/n), +∞[×Mn (R) to Mn (R) which takes the pair (ω, A) to T (ω, A) ≡ (ω − 1)(trA)In + (A + At ) . Here Mn (R) denotes the space of n × n matrices with real entries, In denotes the n × n identity matrix, trA and At denote the trace and the transpose matrix of A, respectively. We observe that (ω − 1) plays the role of the ratio between the first and second Lam´e constants and that the classical linearization of the Piola Kirchoff tensor equals the second Lam´e constant times T (ω, ·) (cf., e.g., Kupradze, Gegelia, Bashele˘ıshvili, and Burchuladze [21]). We also note that div T (ω, Du) = ∆u + ω∇div u , for all regular vector valued functions u. Now let G be a (nonlinear) function from ∂Ωh × Rn to Rn , let B ∈ Mn (R), and let  ∈]0, 0 [. We are in the position to introduce the following nonlinear traction boundary value problem  in S[Ωhp, ]− ,  div T (ω, Du) = 0 h − u(x + qej ) = u(x) + Bej (1.3) p, ] , ∀j ∈ {1, . . . , n},
∀x ∈ clS[Ω  h T (ω, Du(x))νΩhp, (x) = G (x − p)/, u(x) ∀x ∈ ∂Ωp, , where νΩhp, denotes the outward unit normal to ∂Ωhp, . Because of the presence of a nonlinear term in the third equation of problem (1.3), we cannot claim in general the existence of a solution. However, we know by [16] that under suitable assumptions there exists 1 ∈]0, 0 ] such that the boundary value problem in m,α (1.3) has a solution u(, ·) in Cloc (clS[Ωhp, ]− , Rn ) for all  ∈]0, 1 [. Moreover, the family {u(, ·)}∈]0,1 [ is uniquely determined (for  small) by its limiting behavior as  tends to 0 and the dependence of u(, ·) upon the parameter  can be described in terms of real analytic maps of  defined in an open neighborhood of 0 (see also Theorems 2.3 and 2.8). For the definition and properties of real analytic maps, we refer, e.g., to Deimling [17, §15]. In this paper, we are interested in analyzing the behavior of the energy integral Z 
1

t  E ω, u(, ·) ≡ tr T ω, Dx u(, x) Dx u(, x) dx 2 Q\clΩhp, as  approaches 0. By exploiting the representation results for u(, ·) obtained in [16] (see also Theorem 2.3),
we prove in Theorem 3.1 that E ω, u(, ·) equals for  small and positive an analytic function of 
defined in a whole neighborhood of 0. We observe that such result implies in particular that E ω, u(, ·) can be expressed as a converging power series of the variable . The functional analytic approach adopted in this paper has been previously exploited by Lanza de Cristoforis and the authors to analyze singular perturbation problems for the Laplace operator in a bounded 2

domain with a small hole in [14, 22, 23, 24, 25]. Later, it has been extended to problems related to the system of equations of the linearized elasticity in [10, 11, 12, 13] and to the Stokes system in [8, 9]. Concerning problems in an infinite periodically perforated domain, we mention [15, 16, 26, 29, 30]. Singularly perturbed boundary value problems have been largely investigated with the methods of asymptotic analysis. As an example, we mention the works of Bonnaillie-No¨el, Dambrine, Tordeux, and Vial [6], Kozlov, Maz’ya, and Movchan [20], Maz’ya, Movchan, and Nieves [27], Maz’ya, Nazarov, and Plamenewskij [28], Nazarov and Sokolowski [31], Ozawa [32], Ward and Keller [37]. In particular, in connection with periodic problems, we mention Ammari, Garapon, Kang, and Lee [1], Ammari, Kang, and Kim [2], Ammari, Kang, and Lim [3], and Ammari, Kang, and Touibi [4]. Moreover, boundary value problems in multiply connected periodic domains have been analyzed with the method of functional equations. Here we mention, e.g., Castro, Pesetskaya, and Rogosin [7], Dryga´s and Mityushev [18], Rogosin, Dubatovskaya, and Pesetskaya [34]. The paper is organized as follows. In Section 2, we recall some results of [16] concerning the existence, the local uniqueness, and the behavior of a family of solutions of problem (1.3). In Section 3, we prove a
real analytic representation result for the energy integral E ω, u(, ·) .

2

EXISTENCE AND LOCAL UNIQUENESS OF A PARTICULAR FAMILY OF SOLUTIONS

In this section we recall some results of [16]. We use the following notation. Let m ∈ N\{0}, α ∈]0, 1[. Let Ωh be as in assumption (1.1). If G ∈ C 0 (∂Ωh × Rn , Rn ), then we denote by FG the (nonlinear nonautonomous) composition operator from C 0 (∂Ωh , Rn ) to itself which takes v ∈ C 0 (∂Ωh , Rn ) to the function FG [v] from ∂Ωh to Rn defined by FG [v](t) ≡ G(t, v(t)) ∀t ∈ ∂Ωh . Then we consider the following assumptions. G ∈ C 0 (∂Ωh × Rn , Rn ) . FG maps C

m−1,α

h

(2.1) n

(∂Ω , R ) to itself.

(2.2)

We also note here that if G ∈ C 0 (∂Ωh × Rn , Rn ) is such that FG is real analytic from C m−1,α (∂Ωh , Rn ) to itself, then the gradient matrix Du G(·, ·) of G(·, ·) with respect to the variable in Rn exists. If v˜ ∈ C m−1,α (∂Ωh , Rn ) and dFG [˜ v ] denotes the Fr´echet differential of FG at v˜, then we have dFG [˜ v ](v) =

n X

∀v ∈ C m−1,α (∂Ωh , Rn )

F∂ul G [˜ v ]vl

l=1

(cf. Lanza de Cristoforis [23, Prop. 6.3]). Moreover, Du G(·, ξ) ∈ C m−1,α (∂Ωh , Mn (R))

∀ξ ∈ Rn ,

where C m−1,α (∂Ωh , Mn (R)) denotes the space of functions of class C m−1,α from ∂Ωh to Mn (R). The following theorem ensures, under suitable assumptions, the existence of a particular family of solutions of (1.3) and clarifies its behavior for  small and positive (see [16, Thms. 5.5, 6.1 and Def. 5.6]). Theorem 2.3. Let ω ∈]1 − (2/n), +∞[. Let m ∈ N \ {0}, α ∈]0, 1[. Let Ωh be as in assumption (1.1). Let p ∈ Q. Let 0 be as in assumption (1.2). Let B ∈ Mn (R). Let G be as in assumptions (2.1), (2.2). Assume that FG is real analytic from C m−1,α (∂Ωh , Rn ) to itself. (2.4) n ˜ Assume that there exists ξ ∈ R such that Z Z ˜ dσt = 0 and det ˜ dσt 6= 0. G(t, ξ) Du G(t, ξ) (2.5) ∂Ωh

∂Ωh

m,α Then there exist 1 ∈]0, 0 [ and a family {u(, ·)}∈]0,1 [ such that u(, ·) belongs to Cloc (clS[Ωhp, ]− , Rn ) and solves problem (1.3) for all  ∈]0, 1 [. Moreover, the family {u(, ·)}∈]0,1 [ satisfies the following conditions.

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˜ be a bounded open subset of Rn such that clΩ ˜ ⊆ Rn \ (p + qZn ). Let k ∈ N. Then there exist (i) Let Ω ˜ Rn ) such that ˜Ω˜ ∈]0, 1 ] and a real analytic map UΩ˜ from ] − ˜Ω˜ , ˜Ω˜ [ to C k (clΩ, ˜ ⊆ S[Ωh ]− clΩ p,

∀ ∈] − ˜Ω˜ , ˜Ω˜ [ ,

and that ˜, ∀x ∈ clΩ

u(, x) = UΩ˜ [](x)

∀ ∈]0, ˜Ω˜ [ .

Moreover, UΩ˜ [0](x) = Bq −1 (x − p) + ξ˜

˜. ∀x ∈ clΩ

(2.6)

˜ r be a bounded open subset of Rn \ clΩh . Then there exist ˜ ˜ ∈]0, 1 ] and a real analytic map (ii) Let Ω r,Ωr ˜ r , Rn ) such that Ur,Ω˜ r from ] − ˜r,Ω˜ r , ˜r,Ω˜ r [ to C m,α (clΩ ˜ r ⊆ Q \ Ωhp, p + clΩ

∀ ∈] − ˜r,Ω˜ r , ˜r,Ω˜ r [\{0} ,

and that ˜r , ∀t ∈ clΩ

u(, p + t) = Ur,Ω˜ r [](t)

∀ ∈]0, ˜r,Ω˜ r [ .

Moreover, Ur,Ω˜ r [0](t) = ξ˜

˜r . ∀t ∈ clΩ

(2.7)

(Here the letter ‘r’ stands for ‘rescaled’.) Sufficient conditions can be exhibited for the assumption in (2.4), as observed in Lanza de Cristoforis ˜ from clΩh × Rn to Rn is such that the map F ˜ which takes v ∈ [23, p. 972]. Indeed, if a function G G ˜ v(t)) is real analytic from C m−1,α (clΩh , Rn ) to itself, then F ˜ C m−1,α (clΩh , Rn ) to FG˜ [v](t) ≡ G(t, G h n |∂Ω ×R

maps real analytically C m−1,α (∂Ωh , Rn ) to itself, as can be readily deduced by the existence of a linear and continuous extension operator from C m−1,α (∂Ωh , Rn ) to C m−1,α (clΩh , Rn ) (see Gilbarg and Trudinger [19, Lemmas 6.37, 6.38] and Troianiello [35, Lemma 1.5, p. 16]). Then a sufficient condition on a function ˜ ∈ C ∞ (clΩh × Rn , Rn ) so that F ˜ is real analytic from C m−1,α (clΩh , Rn ) to itself is that the functions G G ˜ ξ), with |β| ≤ sup(1, m − 1) are analytic in ξ uniformly with respect to t (see Valent [36, (t, ξ) 7→ Dβ G(t, p. 38 and Thm. 5.2, p. 44]). On the other hand, if, for example, the function G can be continued real analytically in an open neighborhood of ∂Ωh × Rn in R2n , then one can deduce the real analyticity of FG from C m−1,α (∂Ωh , Rn ) to itself, by exploiting results concerning the analyticity of the autonomous composition operator such as those of B¨ ohme and Tomi [5, p. 10] or of Preciso [33, Prop. 1.1, p. 101]. In the following theorem, we present a local uniqueness result for the family of solutions {u(, ·)}∈]0,1 [ (see [16, Thm. 7.1]). Theorem 2.8. Let ω ∈]1 − (2/n), +∞[. Let m ∈ N \ {0}, α ∈]0, 1[. Let Ωh be as in assumption (1.1). Let p ∈ Q. Let 0 be as in assumption (1.2). Let B ∈ Mn (R). Let G be as in assumptions (2.1), (2.2), (2.4). Let ξ˜ ∈ Rn . Let assumption (2.5) hold. Let 1 and {u(, ·)}∈]0,1 [ be as in Theorem 2.3. Let {εj }j∈N be a sequence in ]0, 0 [ converging to 0. Let {uj }j∈N be a sequence of functions such that m,α uj ∈ Cloc (clS[Ωhp,εj ]− , Rn )

∀j ∈ N ,

uj solves problem (1.3) with  ≡ εj ∀j ∈ N , lim uj (p + εj ·)|∂Ωh = ξ˜ in C m−1,α (∂Ωh , Rn ) . j→∞

Then there exists j0 ∈ N such that uj = u(εj , ·)

∀j ∈ N such that j ≥ j0 .

(Here uj (p + εj ·)|∂Ωh denotes the map from ∂Ωh to Rn which takes t to uj (p + εj t).)

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3

A FUNCTIONAL ANALYTIC REPRESENTATION THEOREM FOR THE ENERGY INTEGRAL OF THE SOLUTION

In this section we consider the energy integral of our solution u(, ·) and prove the following statement. Theorem 3.1. Let ω ∈]1 − (2/n), +∞[. Let m ∈ N \ {0}, α ∈]0, 1[. Let Ωh be as in assumption (1.1). Let p ∈ Q. Let 0 be as in assumption (1.2). Let B ∈ Mn (R). Let G be as in assumptions (2.1), (2.2), (2.4). Let ξ˜ ∈ Rn . Let assumption (2.5) hold. Let 1 and {u(, ·)}∈]0,1 [ be as in Theorem 2.3. Then there exist ˜ ∈]0, 1 ] and a real analytic function F from ] − ˜, ˜[ to R such that
E ω, u(, ·) = F[] ∀ ∈]0, ˜[ . (3.2) Moreover, n

t  1 Y   qjj tr T ω, Bq −1 Bq −1 . 2 j=1

F[0] =

(3.3)

Proof. If  ∈]0, 1 [, by applying the Divergence Theorem in the domain Q \ clΩhp, , we obtain Z
t
1 u(, x) T (ω, Dx u(, x))νQ (x) dσx E ω, u(, ·) = 2 ∂Q Z
t 1 − u(, x) T (ω, Dx u(, x))νΩhp, (x) dσx 2 ∂Ωhp, Z
t 1 u(, x) T (ω, Dx u(, x))νQ (x) dσx = 2 ∂Q Z
t n−1 u(, p + s) T (ω, Dx u(, p + s))νΩh (s) dσs . − 2 ∂Ωh Here, νQ and νΩh denote the outward unit normal to ∂Q and to ∂Ωh , respectively. Then we fix R ∈]0, +∞[ ˜ r ≡ Bn (0, R) \ clΩh , and we take ˜ ˜ and U ˜ as in Theorem such that clΩh ⊆ Bn (0, R). Next we set Ω r,Ωr r,Ωr 2.3 (ii). A simple computation shows that Z
t n−1 − u(, p + s) T (ω, Dx u(, p + s))νΩh (s) dσs 2 ∂Ωh Z
t n−2 =− Ur,Ω˜ r [](s) T (ω, Ds Ur,Ω˜ r [](s))νΩh (s) dσs , 2 ∂Ωh for all  ∈]0, ˜r,Ω˜ r [. Accordingly, we set n−2 F1 [] ≡ − 2

Z ∂Ωh


t Ur,Ω˜ r [](s) T (ω, Ds Ur,Ω˜ r [](s))νΩh (s) dσs

for all  ∈] − ˜r,Ω˜ r , ˜r,Ω˜ r [. We note that Theorem 2.3 (ii) ensures that F1 is a real analytic function from ] − ˜r,Ω˜ r , ˜r,Ω˜ r [ to R. Then, by equality (2.7), we have Z δ2,n ˜ Ωh (s) dσs = 0 , ξ˜t T (ω, Ds ξ)ν F1 [0] = − 2 ∂Ωh ˜ of Rn of class C ∞ such where δ2,n = 1 if n = 2 and δ2,n = 0 if n ≥ 3. Now we fix a bounded open subset Ω n n ˜ ⊆ R \ (p + qZ ) and that ∂Q ⊆ Ω. ˜ Then we take ˜˜ and U ˜ as in Theorem 2.3 (i). Clearly, we that clΩ Ω Ω have Z
t 1 u(, x) T (ω, Dx u(, x))νQ (x) dσx 2 ∂Q Z
t 1 = UΩ˜ [](x) T (ω, Dx UΩ˜ [](x))νQ (x) dσx , 2 ∂Q 5

for all  ∈]0, ˜Ω˜ [. Accordingly, we set 1 F2 [] ≡ 2

Z ∂Q


t UΩ˜ [](x) T (ω, Dx UΩ˜ [](x))νQ (x) dσx

for all  ∈] − ˜Ω˜ , ˜Ω˜ [. We note that Theorem 2.3 (i) ensures that F2 is a real analytic function from ] − ˜Ω˜ , ˜Ω˜ [ to R. Then, by equality (2.6) and by the Divergence Theorem, we have Z
t 1 ˜ Q (x) dσx Bq −1 (x − p) + ξ˜ T (ω, Dx (Bq −1 (x − p) + ξ))ν F2 [0] = 2 ∂Q Z 

 1 ˜ Dx (Bq −1 (x − p) + ξ) ˜ t dx = tr T ω, Dx (Bq −1 (x − p) + ξ) 2 Q n

t  1 Y   = qjj tr T ω, Bq −1 Bq −1 . 2 j=1 As a consequence, if we set ˜ ≡ min{˜ r,Ω˜ r , ˜Ω˜ } and F[] ≡ F1 [] + F2 []

∀ ∈] − ˜, ˜[ ,

we deduce that F is a real analytic function from ] − ˜, ˜[ to R such that equalities (3.2) and (3.3) hold. Thus the proof is complete. 2
Remark 3.4. By Theorem 3.1, E ω, u(, ·) can be continued real analytically for  ≤ 0 and thus can be represented for  small and positive in terms of a power series of the variable  which converges absolutely on a whole neighborhood of 0.

References [1] H. Ammari, P. Garapon, H. Kang, and H. Lee, Effective viscosity properties of dilute suspensions of arbitrarily shaped particles, Asymptot. Anal., 80(2012), 189 – 211. [2] H. Ammari, H. Kang, and K. Kim, Polarization tensors and effective properties of anisotropic composite materials, J. Differential Equations, 215(2005), 401 – 428. [3] H. Ammari, H. Kang, and M. Lim, Effective parameters of elastic composites, Indiana Univ. Math. J., 55(2006), 903 – 922. [4] H. Ammari, H. Kang, and K. Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptot. Anal., 41(2005), 119 – 140. [5] R. B¨ ohme and F. Tomi, Zur Struktur der L¨osungsmenge des Plateauproblems, Math. Z., 133(1973), 1 – 29. [6] 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), 1853 – 1882. [7] 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), 1085 – 1100. [8] M. Dalla Riva, Energy integral of the Stokes flow in a singularly perturbed exterior domain, Opuscula Math., 32(2012), 647 – 659. [9] M. Dalla Riva, Stokes flow in a singularly perturbed exterior domain, Complex Var. Elliptic Equ., 58(2013), 231 – 257. [10] M. Dalla Riva and M. Lanza de Cristoforis, A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach, Analysis (Munich), 30(2010), 67 – 92.

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[11] M. Dalla Riva and M. Lanza de Cristoforis, Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach, Complex Var. Elliptic Equ., 55(2010), 771 – 794. [12] M. Dalla Riva and M. Lanza de Cristoforis, Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach, Eurasian Math. J., 1(2010), 31 – 58. [13] 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), 811 – 833. [14] M. Dalla Riva and P. Musolino, Real analytic families of harmonic functions in a domain with a small hole, J. Differential Equations, 252(2012), 6337 – 6355. [15] 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), 24 – 46. [16] M. Dalla Riva and P. Musolino, A singularly perturbed nonlinear traction problem in a periodically perforated domain: a functional analytic approach, Math. Methods Appl. Sci., 37(2014), 106 – 122. [17] K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, 1985. [18] P. Dryga´s and V. Mityushev, Effective conductivity of unidirectional cylinders with interfacial resistance, Quart. J. Mech. Appl. Math., 62(2009), 235 – 262. [19] D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order, vol. 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, second ed., 1983. [20] V. Kozlov, V. Maz’ya, and A. Movchan, Asymptotic analysis of fields in multi-structures, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1999. [21] V.D. Kupradze, T.G. Gegelia, M.O. Bashele˘ıshvili, T.V. Burchuladze, Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity, North-Holland Series in Applied Mathematics and Mechanics, vol. 25, North-Holland Publishing Co., Amsterdam, 1979. [22] 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), 1 – 27. [23] 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), 945 – 977. [24] M. Lanza de Cristoforis, Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Analysis (Munich), 28(2008), 63 – 93, [25] M. Lanza de Cristoforis, Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach, Complex Var. Elliptic Equ., 55(2010), 269 – 303. [26] 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), 511 – 536. [27] V. Maz’ya, A. Movchan, and M. Nieves, Green’s kernels and meso-scale approximations in perforated domains, Lecture Notes in Mathematics, vol. 2077, Springer, Berlin, 2013. [28] V. Maz’ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vols. I, II, vols. 111, 112 of Operator Theory: Advances and Applications, Birkh¨ auser Verlag, Basel, 2000. [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), 334 – 349. [30] P. Musolino, A singularly perturbed Dirichlet problem for the Poisson equation in a periodically perforated domain. A functional analytic approach. In A. Almeida, L. Castro, F.-O. Speck, editors, Advances in Harmonic Analysis and Operator Theory, The Stefan Samko Anniversary Volume, Operator Theory: Advances and Applications, 229, 269 – 289, Birkh¨auser Verlag, Basel, 2013. 7

[31] S.A. Nazarov and J. Sokolowski, Asymptotic analysis of shape functionals, J. Math. Pures Appl. (9), 82(2003), 125 – 196. [32] S. Ozawa, Electrostatic capacity and eigenvalues of the Laplacian, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30(1983), 53 – 62. [33] L. Preciso, Regularity of the composition and of the inversion operator and perturbation analysis of the conformal sewing problem in Roumieu type spaces, Nat. Acad. Sci. Belarus, Proc. Inst. Math., 5(2000), 99 – 104. [34] S. Rogosin, M. Dubatovskaya, and E. Pesetskaya, Eisenstein sums and functions and their application ˇ at the study of heat conduction in composites, Siauliai Math. Semin., 4(2009), 167 – 187. [35] G.M. Troianiello, Elliptic differential equations and obstacle problems, Plenum Press, New York and London, 1987. [36] T. Valent, Boundary value problems of finite elasticity. Local theorems on existence, uniqueness and analytic dependence on data, Springer-Verlag, New York, 1988. [37] M.J. Ward and J.B. Keller, Strong localized perturbations of eigenvalue problems, SIAM J. Appl. Math., 53(1993), 770 – 798.

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