Harmonic functions in a domain with a small hole. A functional analytic approach M. Dalla Riva

1

P. Musolino

Introduction

In this survey, we present some recent results obtained by the authors on the asymptotic behavior of harmonic functions in a bounded domain with a small hole. Particular attention will be paid to the case of the solutions of a Dirichlet problem for the Laplace operator in a perforated domain. We fix once for all n ∈ N \ {0, 1} ,

α ∈]0, 1[ .

Then we take two open sets Ωi and Ωo in the n-dimensional Euclidean space R . We assume that Ωi and Ωo satisfy the following condition. n

Ωi and Ωo are open bounded connected subsets of Rn of class C 1,α such that Rn \ clΩi and Rn \ clΩo are connected, n

i

(1)

o

and such that the origin 0 of R belongs both to Ω and Ω . Here cl denotes the closure of a set. For the definition of functions and sets of the usual Schauder classes C 0,α and C 1,α , we refer for example to Gilbarg and Trudinger [GiTr01, §6.2]. The set Ωo will represent the unperturbed domain where we shall make a hole. On the other hand, the set Ωi will play the role of the shape of the perforation. Here, the letter ‘i’ stands for ‘inner domain’ and the letter ‘o’ stands for ‘outer domain’. We note that condition (1) implies that Ωi and Ωo have no holes and that there exists a real number 0 such that 0 > 0 and clΩi ⊆ Ωo for all  ∈] − 0 , 0 [. We are now in the position to define the perforated domain Ω(): Ω() ≡ Ωo \ clΩi

∀ ∈] − 0 , 0 [ .

In other words, the set Ω() is obtained by removing from Ωo the closure of the set Ωi , which can be seen as a hole. If  ∈] − 0 , 0 [\{0}, then by a simple topological argument one can see that Ω() is an open bounded connected subset of Rn of class C 1,α . For each  ∈ ] − 0 , 0 [, the boundary ∂Ω() of Ω() consists of the two connected components 1

∂Ωo and ∂Ωi . In particular, ∂Ωo is the ‘outer boundary’ of Ω() and ∂Ωi is the ‘inner boundary’. We also note that Ω(0) = Ωo \ {0}. For each  ∈] − 0 , 0 [\{0} we want to consider a Dirichlet problem for the Laplace operator in the perforated domain Ω(). In order to do so, we fix two functions f i ∈ C 1,α (∂Ωi ) and f o ∈ C 1,α (∂Ωo ), and we define the Dirichlet datum f ∈ C 1,α (∂Ω()) as follows: ( f i (x/) if x ∈ ∂Ωi , f (x) ≡ f o (x) if x ∈ ∂Ωo . Then for each  ∈]−0 , 0 [\{0} we consider the following boundary value problem  ∆u = 0 in Ω() , (2) u(x) = f (x) for x ∈ ∂Ω() . As is well known, the problem in (2) has a unique solution in C 1,α (clΩ()), and we denote such solution by u . Our aim is to investigate the behavior of the solution u of (2) as  tends to 0. We observe that problem (2) is clearly singular when  = 0. Indeed, the domain Ω() is degenerate for  = 0 and also the Dirichlet datum on ∂Ωi does not make sense for  = 0. Therefore, in order to study the behavior of u , we can fix a point which belongs to Ω() for all  that are close to 0, and see what happens to the value of the solution u at this fixed point as  approaches 0. Also, we can choose to approach to the degenerate value  = 0, for example, from positive values of . So assume that p ∈ Ωo \ {0} , (3) and that p ∈]0, 0 [ is such that p ∈ Ω() for all  ∈]0, p [.

(4)

We note that (4) implies that the point p belongs to the domain of the function u for all  ∈]0, p [, and therefore it makes sense to consider the value u (p) for  ∈]0, p [. Thus we can ask the following question. What can be said of the map from ]0, p [ to R which takes  to u (p) when  is close to 0?

(5)

The behavior of the solutions of boundary value problems in domains with small holes has been investigated, for example, with methods of asymptotic analysis. With such approach, one would try to answer to the question in (5) by producing an asymptotic expansion of u (p) for  close to 0. It is impossible to provide a complete list of all the contributions with this method. As an example, we mention the works by Bonnaillie-Noël and Dambrine [BoDa13], Il’in [Il92], Maz’ya, Movchan, and Nieves [MaMoNi13], Maz’ya, Nazarov, and Plamenevskij [MaNaPl00i, MaNaPl00ii]. Moreover, the study of problems of this type has revealed to be a powerful tool in the frame of shape optimization (cf. Novotny and Sokołowsky [NoSo13]). Applications of these investigations 2

for example to inverse problems are widely illustrated in Ammari and Kang [AmKa07] and Ammari, Kang, and Lee [AmKaLe09]. Here instead we wish to characterize the behavior of u at  = 0 by a different approach. For example, we would try to represent u (p) for  > 0 in terms of real analytic functions of the variable  defined on a whole neighborhood of 0, and of possibly singular at  = 0 but explictly known functions of  (such as log , −1 , etc.). Then, if we knew, for example, that u (p) equals for positive values of  a real analytic function of the variable  defined on a whole neighborhood of 0, we would be able to deduce the existence of 0 ∈]0, p [ and of a sequence {cj }∞ j=0 of real numbers such that u (p) =

∞ X

cj j

∀ ∈]0, 0 [ ,

j=0

where the series in the right hand side converges absolutely on ] − 0 , 0 [. As we shall see, this is indeed the case when n ≥ 3 (cf. Theorem 2.1 below). This method has been applied to investigate perturbation problems for the conformal representation and for boundary value problems for the Laplace operator in a bounded domain with a small hole (cf., e.g., Lanza de Cristoforis [La02, La08]). Later on, the approach has been extended to nonlinear traction problems in elastostatics (cf., e.g., [DaLa11]), to the Stokes’ flow (cf., e.g., [Da13]), and to the case of an infinite periodically perforated domain (cf., e.g., [LaMu14]). Moreover, the authors have analyzed the effective properties of dilute composite materials by this technique (see [DaMu13]). Finally, also (regular) domain perturbation problems in spectral theory have been analyzed with this approach (cf., e.g., Buoso and Lamberti [BuLa13], Lamberti and Provenzano [LaPr13]).

2

What happens when  is positive and close to 0?

In the following theorem, we answer question (5) on the behavior of u (p) as  → 0+ , by exploiting the functional analytic approach proposed by Lanza de Cristoforis. We find convenient to denote by u0 the unique function in C 1,α (clΩo ) such that  ∆u0 = 0 in Ωo , (6) o u0 (x) = f (x) for x ∈ ∂Ωo . Theorem 2.1 (Lanza de Cristoforis [La08]) Let p be as in (3). Then the following statements hold. (i) If n = 2, then there exist p as in (4), p < 1, and a real analytic function Up# from ] − p , p [×]1/ log p , −1/ log p [ to R such that u (p) = Up# [, 1/ log ] 3

∀ ∈]0, p [ ,

and that u0 (p) = Up# [0, 0] (ii) If n ≥ 3, then there exist p as in (4) and a real analytic function Up from ] − p , p [ to R such that u (p) = Up []

∀ ∈]0, p [ ,

and that u0 (p) = Up [0] Now, instead of considering the behavior of the value of u at a fixed point, as done in Theorem 2.1, we could consider the restriction of u to the closure of a suitable open subset of Ωo \ {0}. More precisely, we note that if ˜ is a bounded open subset of Ωo such that 0 6∈ clΩ ˜, Ω

(7)

then there exists Ω˜ such that ˜ ∩ clΩi = ∅ for all  ∈] −  ˜ ,  ˜ [ . Ω˜ ∈]0, 0 [ and clΩ Ω Ω

(8)

˜ ⊆ clΩ() for all  ∈] −  ˜ ,  ˜ [. As consequence, if  ∈]0,  ˜ [, In particular, clΩ Ω Ω Ω ˜ Then we describe then it makes sense to consider the restriction of u to clΩ. the behavior of u|clΩ˜ in the following. ˜ be as in (7). Then the Theorem 2.2 (Lanza de Cristoforis [La08]) Let Ω following statements hold. (i) If n = 2, then there exist Ω˜ as in (8), Ω˜ < 1, and a real analytic map ˜ such that UΩ˜# from ] − Ω˜ , Ω˜ [×]1/ log Ω˜ , −1/ log Ω˜ [ to C 1,α (clΩ) u (x) = UΩ˜# [, 1/ log ](x)

˜ , ∀ ∈]0,  ˜ [ , ∀x ∈ clΩ Ω

(9)

and that u0|clΩ˜ = UΩ˜# [0, 0]. (ii) If n ≥ 3, then there exist Ω˜ as in (8) and a real analytic map UΩ˜ from ˜ such that ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) ˜ , ∀ ∈]0,  ˜ [ , ∀x ∈ clΩ Ω

u (x) = UΩ˜ [](x)

(10)

and that u0|clΩ˜ = UΩ˜ [0]. We note that in Theorem 2.2 the real analytic maps UΩ˜# and UΩ˜ have values ˜ in the Banach space C 1,α (clΩ). Here we just recall that if X , Y are (real) Banach spaces and if F is a map from an open subset W of X to Y, then F is real analytic in W if for every x0 ∈ W there exist P r > 0 and continuous j j symmetric j-linear operators A from X to Y such that j j≥1 kAj kr < ∞ and P F (x0 + h) = F (x0 ) + j≥1 Aj (h, . . . , h) for all h ∈ X with khkX ≤ r (cf., e.g., Deimling [De85, p. 150]).

4

Theorem 2.2 has been proved in Lanza de Cristoforis [La08, Theorem 5.3], where also real analyticity properties of the solution upon perturbations of Ωo and Ωi are considered. Furthermore, Theorem 2.2 could also be deduced by some more recent results concerning real analytic families of harmonic functions (cf. [DaMu12, Proposition 4.1] and [DaMu15, Theorem 3.1]). ˜ then the map which takes a function Moreover, we observe that if p ∈ clΩ, ˜ to u(p) is linear and continuous (and thus real analytic). Since u ∈ C 1,α (clΩ) the composition of real analytic maps is real analytic, by Theorem 2.2 we deduce the validity of Theorem 2.1.

3

What happens for  negative?

Now we would like to investigate the validity of equalities (9) and (10) when  is negative. As we have seen, the behavior of u for  close to 0 in case n = 2 and in case n ≥ 3 are different. As a consequence, we need to analyze separately these two cases.

3.1

Case of dimension n ≥ 3

We now observe that both u and UΩ˜ [] in equality (10) are defined also for negative values of . However, by Theorem 2.2, we just know that the equality in (10) holds when  is small and positive. As a consequence, it is natural to ask the following question. Does the equality in (10) hold also for  negative?

(11)

In [DaMu12], it has been shown that the answer to the question in (11) depends on the parity of the dimension n. The following theorem says that if the dimension n is even and bigger than 3 (i.e., n = 4, 6, 8, . . . ), then the equality in (10) holds also for  < 0 (cf. [DaMu12, Theorem 3.1 and Proposition 4.1]). Moreover, if u0 is the solution of problem (6), equality (10) holds in a whole neighborhood of 0, and in particolar also for  = 0. ˜  ˜ be as in (7), (8), respectively. Theorem 3.1 Let n be even and n ≥ 3. Let Ω, Ω ˜ such that Then there exists a real analytic map UΩ˜ from ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) u (x) = UΩ˜ [](x)

˜ , ∀ ∈] −  ˜ ,  ˜ [ . ∀x ∈ clΩ Ω Ω

We now turn to consider case n odd. As we shall see, if n is odd (i.e., n = 3, 5, 7, . . . ), then the validity of the equality in (10) also for  < 0 has to be considered as a very exceptional situation. Indeed, we have the following theorem (cf. [DaMu12, Proposition 4.3]). Theorem 3.2 Let n be odd and n ≥ 3. Then the following statements are equivalent.

5

˜  ˜ as in (7), (8), respectively, and a real analytic map U ˜ (i) There exist Ω, Ω Ω ˜ such that from ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) ˜ , ∀ ∈] −  ˜ ,  ˜ [\{0} . ∀x ∈ clΩ Ω Ω

u (x) = UΩ˜ [](x) (ii) There exists c ∈ R such that f i (x) = c

∀x ∈ ∂Ωi ,

f o (x) = c

∀x ∈ ∂Ωo

(and thus u (x) = c for all x ∈ clΩ() and  ∈] − 0 , 0 [\{0}). ˜  ˜ are as in (7), (8), Clearly, if statement (ii) of Theorem 3.2 holds and Ω, Ω ˜ defined by respectively, then the map UΩ˜ from ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) UΩ˜ [](x) = c

˜, ∀x ∈ clΩ

∀ ∈] − Ω˜ , Ω˜ [ ,

is such that the equality in (10) holds for  ∈] − Ω˜ , Ω˜ [\{0}, and therefore we deduce the validity of statement (i). On the other hand, Theorem 3.2 says in ˜ as in (7) for which we particular that if there exists at least one open subset Ω can find a small positive number Ω˜ as in (8) and a real analytic map UΩ˜ from ˜ such that the equality in (10) holds for  ∈] −  ˜ ,  ˜ [\{0}, ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) Ω Ω then we are in the very exceptional situation that f i and f o are both equal to the same constant c ∈ R (and that accordingly u = c on clΩ() for all  ∈] − 0 , 0 [\{0}). Hence, if n is odd the validity of equality (10) also for  negative has to be considered as a very special situation which happens only in the trivial case in which the functions u for  ∈] − 0 , 0 [\{0} are all equal to the same constant.

3.2

Case of dimension n = 2

We now turn to consider the case of dimension n = 2. Also in this case, we would like to say something about the validity of equality (9) for  < 0. In particular we would like to replace the pair (, 1/ log ), where the map UΩ˜# is evaluated when  > 0, by a convenient pair which makes sense also for  < 0, in a way to preserve the validity of equality (9) for  in a whole neighborhood of 0. We do so in the following theorem (cf. [DaMu15, Theorem 3.1]). ˜  ˜ be as in (7), (8), respectively, with  ˜ < 1. Theorem 3.3 Let n = 2. Let Ω, Ω Ω Then there exist an open neighborhood UΩ˜ of {(, 1/ log ||) :  ∈]−Ω˜ , Ω˜ [\{0}}∪ ˜ such that {(0, 0)} in R2 and a real analytic map UΩ˜# from UΩ˜ to C 1,α (clΩ) u (x) = UΩ˜# [, 1/ log ||](x)

˜ , ∀ ∈] −  ˜ ,  ˜ [\{0} . ∀x ∈ clΩ Ω Ω

Now we would like to consider boundary data f o and f i in such a way to get rid of the logarithmic behavior of u for  small. In other words, we would like that the following condition (a) holds. 6

˜  ˜ as in (7), (8), respectively, there exists a real analytic map (a) For all Ω, Ω ˜ such that VΩ˜ from ] − Ω˜ , Ω˜ [ to C 1,α (clΩ) u (x) = VΩ˜ []

˜ , ∀ ∈] −  ˜ ,  ˜ [ . ∀x ∈ clΩ Ω Ω

In [DaMu15, Theorem 3.6], we show that condition (a) is equivalent to the following condition (b). (b) There exist p, p as in (3), (4), respectively, and a real analytic map Vp from ] − p , p [ to R such that p ∈ Ω() for all  ∈] − p , p [ and u (p) = Vp []

∀ ∈]0, p [ .

This means that either u (p) displays a logarithmic behavior for every point p ∈ Ωo \ {0}, or u (p) does not display a logarithmic behavior for any point p ∈ Ωo \ {0}. Also, by [DaMu15, Theorem 3.6] there exists a pair of functions (ρo [], ρi []) ∈ C 0,α (∂Ωo ) × C 0,α (∂Ωi ) which depends only on , ∂Ωo , and ∂Ωi , such that (a) and (b) are equivalent to the following condition (c). R R (c) It holds ∂Ωo f o ρo [] dσ + ∂Ωi f i ρi [] dσ = 0 for all  ∈] − 0 , 0 [. The advantage of condition (c) with respect to (a) and (b) is that (c) can be verified on the boundary data f o and f i and does not require the knowledge of the solution u of (2). In [DaMu15, §3], we also observe that under some convenient assumptions, condition (c) can become very explicit. For example, if f o and f i are both constant functions, then condition (c) is equivalent to the fact that f o and f i are identically equal to the same real number. If instead both Ωo Rand Ωi coincide the unit ball B2 of R2 , then condition (c) is equivalent R with i o to ∂B2 f dσ = ∂B2 f dσ.

4

Asymptotic expansion of the solution of a Dirichlet problem in a perforated domain (n = 2)

As already mentioned, the functional approach of the authors can be used to compute asymptotic expansions for the solutions of boundary value problems in perforated domains. In particular, the results of [DaMu15] have been exploited in [DaMuRo] to prove the following expansions for the solution u of the Dirichlet problem (2) for the Laplace operator in a bounded planar domain with a small hole. Theorem 4.1 Let n = 2. Then there exist a family {λM,(j,l) }(j,l)∈N2 , l≤j+1 of functions from (clΩo ) \ {0} to R, and a family {λm,(j,l) }(j,l)∈N2 , l≤j+1 of functions from R2 \ Ωi to R, and r0 ∈ R such that the following statements hold. (i) Let ΩM ⊆ Ωo be open and such that 0 ∈ / clΩM . Then there exists 0M ∈ i ]0, 0 ]∩]0, 1[ such that clΩM ∩ clΩ = ∅ for all  ∈] − 0M , 0M [ and such 7

that u|clΩM =

∞ X

j

j=0

for all  ∈] −

0M , 0M [\{0}. ∞ X

j+1 X l=0

λM,(j,l)|clΩM (r0 + (2π)−1 log ||)l

Moreover, the series j

j=0

j+1 X λM,(j,l)|clΩM η l (r0 η + (2π)−1 )l l=0

converges in C 1,α (clΩM ) uniformly for (, η) belonging to the product of intervals ] − 0M , 0M [×]1/ log 0M , −1/ log 0M [. (ii) Let Ωm ⊆ R2 \clΩi be open and bounded. Then there exists 0m ∈]0, 0 ]∩]0, 1[ such that clΩm ⊆ Ωo for all  ∈] − 0m , 0m [ and such that u (·)|clΩm =

∞ X

j

j+1 X

j=0

l=0

λm,(j,l)|clΩm (r0 + (2π)−1 log ||)l

for all  ∈] − 0m , 0m [\{0}. Moreover, the series ∞ X j=0

j

j+1 X λm,(j,l)|clΩm η l (r0 η + (2π)−1 )l l=0

converges in C 1,α (clΩm ) uniformly for (, η) belonging to the product of intervals ] − 0m , 0m [×]1/ log 0m , −1/ log 0m [. Here above, we denote by u ( ·) the rescaled function which takes x ∈ −1 clΩ() to u (x), for all  ∈] − 0 , 0 [\{0}. Moreover, the letter ‘M ’ stands for ‘macroscopic’, while the letter ‘m’ stands for ‘microscopic’. Indeed, in Theorem 4.1 (i) we analyze the behavior of u far from the hole and in Theorem 4.1 (ii) we consider the solution in proximity of the perforation. Finally, we note that in [DaMuRo] explicit formulae for the families of functions {λM,(j,l) }(j,l)∈N2 , l≤j+1 and {λm,(j,l) }(j,l)∈N2 , l≤j+1 are provided.

5

Real analytic families of harmonic functions in a bounded domain with a small hole

The results of Subsections 3.1 and 3.2 can be deduced from those of [DaMu12] and [DaMu15], where we introduce and study real analytic families of harmonic functions, which are not required to be the solutions of any particular boundary value problem. Therefore, in the present section, the functions u are not necessarily solutions of problem (2). Definition 5.1 Let 1 ∈]0, 0 ]. We say that {u }∈]0,1 [ is a right real analytic family of harmonic functions on Ω() if it satisfies the following conditions (a0)(a2). 8

(a0) u ∈ C 1,α (clΩ()) and ∆u = 0 in Ω() for all  ∈]0, 1 [. (a1) Let ΩM ⊆ Ωo be open and such that 0 ∈ / clΩM . Let M ∈]0, 1 ] be such that clΩM ∩ clΩi = ∅ for all  ∈] − M , M [. Then there exists a real analytic map UM from ] − M , M [ to C 1,α (clΩM ) such that ∀ ∈]0, M [ .

u|clΩM = UM []

(a2) Let Ωm ⊆ Rn \ clΩi be open and bounded. Let m ∈]0, 1 ] be such that clΩm ⊆ Ωo for all  ∈] − m , m [. Then there exists a real analytic map Um from ] − m , m [ to C 1,α (clΩm ) such that u ( · )|clΩm = Um []

∀ ∈]0, m [ .

Definition 5.2 Let 1 ∈]0, 0 ]. We say that {v }∈]−1 ,1 [ is a real analytic family of harmonic functions on Ω() if it satisfies the following conditions (b0)– (b2). (b0) v0 ∈ C 1,α (clΩo ) and ∆v0 = 0 in Ωo , v ∈ C 1,α (clΩ()) and ∆v = 0 in Ω() for all  ∈] − 1 , 1 [\{0}. (b1) Let ΩM ⊆ Ωo be open and such that 0 ∈ / clΩM . Let M ∈]0, 1 ] be such that clΩM ∩ clΩi = ∅ for all  ∈] − M , M [. Then there exists a real analytic map VM from ] − M , M [ to C 1,α (clΩM ) such that v|clΩM = VM []

∀ ∈] − M , M [ .

(b2) Let Ωm ⊆ Rn \ clΩi be an open and bounded subset. Let m ∈]0, 1 ] be such that clΩm ⊆ Ωo for all  ∈] − m , m [. Then there exists a real analytic map Vm from ] − m , m [ to C 1,α (clΩm ) such that v ( · )|clΩm = Vm []

∀ ∈] − m , m [\{0} .

Definition 5.3 Let 1 ∈]0, 0 ]. We say that {w }∈]−1 ,1 [ is a real analytic family of harmonic functions on Ωo if it satisfies the following conditions (c0), (c1). (c0) w ∈ C 1,α (clΩo ) and ∆w = 0 in Ωo for all  ∈] − 1 , 1 [. (c1) The map from ] − 1 , 1 [ to C 1,α (clΩo ) which takes  to w is real analytic. Then we have the validity of the following theorem (cf. [DaMu12] and [DaMu15]). Theorem 5.4 The following statements hold. (i) If the dimension n is even and {u }∈]0,1 [ is a right real analytic family of harmonic functions on Ω(), then there exists a real analytic family of harmonic functions {v }∈]−1 ,1 [ on Ω() such that u = v for all  ∈]0, 1 [. 9

(ii) If the dimension n is odd and {v }∈]−1 ,1 [ is a real analytic family of harmonic functions on Ω(), then there exists a real analytic family of harmonic functions {w }∈]−1 ,1 [ on Ωo such that v = w|clΩ() for all  ∈] − 1 , 1 [. In particular we note that for n odd Theorem 5.4 (ii) implies that for each  ∈] − 1 , 1 [ the function v can be extended inside the hole Ωi to an harmonic function defined on the whole of Ωo .

Acknowledgment The research of M. Dalla Riva was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT– Fundação para a Ciência e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014. The research of M. Dalla Riva was also supported by the Portuguese Foundation for Science and Technology (“FCT–Fundação para a Ciência e a Tecnologia”) with the research grant SFRH/BPD/ 64437/2009. P. Musolino is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The work of M. Dalla Riva and P. Musolino is also supported by “Progetto di Ateneo: Singular perturbation problems for differential operators – CPDA120171/12” of the University of Padova.

References [AmKa07]

Ammari, H., Kang, H.: Polarization and moment tensors. With applications to inverse problems and effective medium theory. Applied Mathematical Sciences, 162. Springer, New York (2007)

[AmKaLe09]

Ammari, H., Kang, H., Lee, H.: Layer potential techniques in spectral analysis. Mathematical Surveys and Monographs, 153. American Mathematical Society, Providence, RI (2009)

[BoDa13]

Bonnaillie-Noël, V., Dambrine, M.: Interactions between moderately close circular inclusions: the Dirichlet-Laplace equation in the plane. Asymptot. Anal. 84, 197–227 (2013)

[BuLa13]

Buoso, D., Lamberti, P.D.: Eigenvalues of polyharmonic operators on variable domains. ESAIM Control Optim. Calc. Var. 19, 1225–1235 (2013)

[De85]

Deimling, K.: Nonlinear functional analysis, Springer Verlag, Berlin (1985)

10

[Da13]

Dalla Riva, M.: Stokes flow in a singularly perturbed exterior domain. Complex Var. Elliptic Equ. 58, 231–257 (2013)

[DaLa11]

Dalla Riva, M., Lanza de Cristoforis, M.: Weakly singular and microscopically hypersingular load perturbation for a nonlinear traction boundary value problem: a functional analytic approach. Complex Anal. Oper. Theory 5, 811–833 (2011)

[DaMuRo]

Dalla Riva, M., Musolino, P., Rogosin, S.V.: Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole. Submitted.

[DaMu12]

Dalla Riva, M., Musolino, P.: Real analytic families of harmonic functions in a domain with a small hole. J. Differential Equations 252, 6337–6355 (2012)

[DaMu13]

Dalla Riva, M., Musolino, P.: A singularly perturbed nonideal transmission problem and application to the effective conductivity of a periodic composite. SIAM J. Appl. Math. 73, 24–46 (2013)

[DaMu15]

Dalla Riva, M., Musolino, P.: Real analytic families of harmonic functions in a planar domain with a small hole. J. Math. Anal. Appl. 442, 37–55 (2015)

[GiTr01]

Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics, SpringerVerlag, Berlin (2001)

[Il92]

Il’in, A.M.: Matching of asymptotic expansions of solutions of boundary value problems. Translations of Mathematical Monographs 102, American Mathematical Society, Providence (1992)

[LaPr13]

Lamberti, P.D., Provenzano, L.: Eigenvalues of polyharmonic operators on variable domains. Eurasian Math. J. 4, 70–83 (2013)

[La02]

Lanza de Cristoforis, M.: Asymptotic behaviour of the conformal representation of a Jordan domain with a small hole in Schauder spaces. Comput. Methods Funct. Theory 2, 1–27 (2003)

[La08]

Lanza de Cristoforis, M.: Asymptotic behaviour 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, 63–93 (2008)

11

[LaMu14]

Lanza de Cristoforis, M., Musolino, P.: A quasi-linear heat transmission problem in a periodic two-phase dilute composite. A functional analytic approach. Commun. Pure Appl. Anal. 13, 2509–2542 (2014)

[MaMoNi13]

Maz’ya, V., Movchan, A., Nieves, M.: Green’s kernels and meso-scale approximations in perforated domains. Lecture Notes in Mathematics 2077, Springer, Berlin (2013)

[MaNaPl00i]

Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I. Volume 111 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel (2000)

[MaNaPl00ii]

Maz’ya, V., Nazarov, S., Plamenevskij, B.: Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II. Volume 112 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel (2000)

[NoSo13]

Novotny, A.A.; Sokołowski, J.: Topological derivatives in shape optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013)

12