The Dirichlet problem in a planar domain with two moderately close holes
M. Dalla Riva ,† and P. Musolino
Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa, Oklahoma 74104, USA.Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, UK.
( )
Abstract: We investigate a Dirichlet problem for the Laplace equation in a domain of R2 with two small close holes. The domain is obtained by making in a bounded open set two perforations at distance ∣ϵ1∣ one from the other and each one of size ∣ϵ1ϵ2∣. In such a domain, we introduce a Dirichlet problem and we denote by uϵ1,ϵ2 its solution. We show that the dependence of uϵ1,ϵ2 upon (ϵ1,ϵ2) can be described in terms of real analytic maps of the pair (ϵ1,ϵ2) defined in an open neighborhood of (0,0) and of logarithmic functions of ϵ1 and ϵ2. Then we study the asymptotic behaviour of of uϵ1,ϵ2 as ϵ1 and ϵ2 tend to zero. We show that the first two terms of an asymptotic approximation can be computed only if we introduce a suitable relation between ϵ1 and ϵ2.
Keywords: Dirichlet problem; singularly perturbed perforated planar domain; moderately close holes; Laplace operator; real analytic continuation in Banach space; asymptotic expansion
2010 Mathematics Subject Classification: 35J25; 31B10; 45A05; 35B25; 35C20
1 Introduction
The asymptotic analysis of elliptic boundary value problems in domains with many holes which collapse one to the other while shrinking their sizes is a topic of growing interest and several authors have recently proposed different techniques and points of view. We mention for example the method based on multiscale asymptotic expansions which have been used by Bonnaillie-Noël, Dambrine, Tordeux, and Vial [5, 6], Bonnaillie-Noël and Dambrine [3], and Bonnaillie-Noël, Dambrine, and Lacave [4] to study problems with two moderately close holes, i.e., problems with two holes whose mutual distance tends to zero while their size tends to zero at faster speed. The case when the number of holes is large has been considered by Maz’ya, Movchan, and Nieves in a series of papers where they propose a mesoscale approximation method to analyse problems for the Laplace operator and for the system of linear elasticity. We mention, for example, Maz’ya and Movchan [23, 24], and Maz’ya, Movchan, and Nieves [25, 26, 27, 28]. The mesoscale approximation method does not require any periodicity assumption. If instead the holes have a periodic structure, then one can resort to the large literature in homogenization theory, where, rather then aiming at obtaining asymptotic expansions, one typically characterizes the limit value of the solution of a perturbed problem as the solution of a limiting problem. We refer, for instance, to the seminal works of Bakhvalov and Panasenko [2], Cioranescu and Murat [8, 9], and Marčenko and Khruslov [22] and to the more recent ‘periodic unfolding method’ used, e.g., by Cioranescu, Damlamian, Donato, Griso, and Zaki [7]).
In this paper, we consider a Dirichlet problem for the Laplace equation in a planar domain with two small close holes. The method adopted is different from those mentioned above. Indeed, we follow the ‘functional analytic approach’ which has been proposed by Lanza de Cristoforis for the analysis of linear and nonlinear singular perturbation problems (see, e.g., Lanza de Cristoforis [17, 19, 20]) and which allows the representation of the solution in terms of elementary functions and of real analytic maps of the singular perturbation parameters. One of the advantages of the method is that real analytic maps can be expanded into power series and thus, as a byproduct of our analysis, we can deduce fully justified asymptotic expansions for the solution with any order of approximation. Moreover, the coefficients of such expansions can be explicitly and constructively computed by solving certain systems of integral equations (as shown in [13]). This method has been exploited for the analysis of Laplace and Poisson problems in domains with small close holes in [11] and in [12], respectively. In both of these papers, the conditions on the boundaries of the holes are of Neumann type. Here, instead, we will study a problem with Dirichlet conditions and we will focus on the two-dimensional case. This case is more involved than the higher dimensional case or the Neumann condition case because of the logarithmic behaviour induced by the two-dimensional fundamental solution. As we shall see, such logarithmic behaviour will force the introduction of a specific relation between the size and the distance of the holes if we wish to pass from the representation of the solution in terms of analytic maps to the explicit computation of the first asymptotic approximation terms.
We now proceed to introduce our problem and we start by defining the geometric setting. We fix once for all a real number α∈]0,1[ and three sets Ωo, Ω1 and Ω2 that satisfy the following condition:
[TABLE]
Here the letter ‘o’ stands for ‘outer domain’ and Ωo will play the role of the unperturbed outer domain in which we make two holes. To do so, we take two points
[TABLE]
and we assume that there exists
[TABLE]
such that
[TABLE]
Here and in the sequel ‘cl’ denotes the closure. Then we define the rescaled sets
[TABLE]
which will play the role of the holes. We observe that, for ϵ1,ϵ2∈R∖{0} and i∈{1,2}, each Ωi(ϵ1,ϵ2) is an open bounded subset of R2 which contains the point ϵ1pi. Instead, when ϵ1=0 or ϵ2=0, Ωi(ϵ1,ϵ2) collapses to a point and we have Ωi(0,ϵ2)={0} and Ωi(ϵ1,0)={ϵ1pi}. In addition, condition (1) implies that
[TABLE]
Then, one sees that
the mutual distance between Ω1(ϵ1,ϵ2) and Ω2(ϵ1,ϵ2) is controlled by ∣ϵ1∣, while their size is proportional to ∣ϵ1ϵ2∣. As a consequence, when both ϵ1 and ϵ2 approach zero, the size tends to zero at a faster rate than the distance. When this happens, one says that the holes are ‘moderately close’. In this paper, we will also consider the case when the size and the distance are comparable, i.e. when ϵ1 tends to zero and ϵ2 stays away from zero.
Since we want the holes to be contained in Ωo, we have to restrict the set of the ‘admissible’ parameters ϵ1 for which we define the perforated domain. Then we take
[TABLE]
such that
[TABLE]
and we consider the pairs (ϵ1,ϵ2) in the rectangular domain [−δ1,δ1]×[−δ2,δ2] as admissible parameters for which we define the perforated domain
[TABLE]
We observe that for ϵ1∈[−δ1,δ1]∖{0} and ϵ2∈[−δ2,δ2]∖{0}, Ω(ϵ1,ϵ2) is an open bounded connected subset of R2 of class C1,α and the boundary of Ω(ϵ1,ϵ2) consists of three connected components: ∂Ωo, ∂Ω1(ϵ1,ϵ2), and ∂Ω2(ϵ1,ϵ2). For ϵ1=0 the set Ω(0,ϵ2) equals Ωo∖{0} and for ϵ2=0 we have Ω(ϵ1,0)=Ωo∖({ϵ1p1}∪{ϵ1p2}). We also find convenient to introduce the notation
[TABLE]
Now that the geometric configuration is settled, we turn to specify the boundary value problem. In order to define the Dirichlet data on ∂Ω(ϵ1,ϵ2), we fix three functions
[TABLE]
Then, for ϵ1∈[−δ1,δ1]∖{0} and ϵ2∈[−δ2,δ2]∖{0}, we consider the following boundary value problem for a function u∈C1,α(clΩ(ϵ1,ϵ2)):
[TABLE]
As is well known the solution of problem (3) exists and is unique. We denote such solution by uϵ1,ϵ2. Our aim is twofold: first, we want to investigate the dependence of uϵ1,ϵ2 upon ϵ1 and ϵ2; then, we want to obtain asymptotic approximations of of uϵ1,ϵ2 as (ϵ1,ϵ2) tends to a degenerate value (0,γ0), with γ0∈[0,δ2[. We will not consider the case when ϵ2 tends to zero and ϵ1 tends to a non zero value, which corresponds to the situation when the holes shrink to two distinct points. Such latter case has been largely investigated in literature (cf., e.g., Maz’ya, Nazarov, and Plamenevskij [29]).
Concerning the first of the two goals, in Theorem 6.6 we provide a representation (of suitable restrictions) of uϵ1,ϵ2 and of the rescaled functions uϵ1,ϵ2(ϵ1p1+ϵ1ϵ2⋅) and uϵ1,ϵ2(ϵ1p2+ϵ1ϵ2⋅) in terms of real analytic functions of the pair (ϵ1,ϵ2) and of the explicitly known functions log∣ϵ1∣ and log∣ϵ1ϵ2∣. The rescaled functions uϵ1,ϵ2(ϵ1ph+ϵ1ϵ2⋅), with h∈{1,2}, describe the solution in proximity of the boundary of the holes and play an important role if one wants to compute quantities related to the solution, such as the energy integral. As a consequence of Theorem 6.6 we see that, for x∈Ωo∖{0} fixed and possibly shrinking δ1 and δ2, we have
[TABLE]
for all ϵ1∈]−δ1,δ1[∖{0} and all ϵ2∈]−δ2,δ2[∖{0}, where uo is the solution of the unperturbed Dirichlet problem in Ωo with boundary datum fo, the functions Ux, F, and Vx are real analytic from ]−δ1,δ1[×]−δ2,δ2[ to R, and Λ(ϵ1,ϵ2) is a 2×2 matrix such that
[TABLE]
with R real analytic from ]−δ1,δ1[×]−δ2,δ2[ to the space of 2×2 real matrices. As we shall see, Λ(ϵ1,ϵ2) is invertible if both ϵ1 and ϵ2 are not zero.
Then, if we want to exploit (4) to deduce asymptotic approximations of the solution as the pair (ϵ1,ϵ2) approaches a degenerate value (0,γ0), we have to compute the inverse of the matrix Λ(ϵ1,ϵ2). If we do so, we obtain an expression which involves the quotient
[TABLE]
(cf. Proposition 7.1). However, the limit of (5) as (ϵ1,ϵ2)→(0,γ0) does not exist when γ0=0.
To overcome this difficulty, we introduce a relation between the parameters ϵ1 and ϵ2: we replace ϵ1 by a positive parameter t and we take ϵ2=γ(t), with γ a function from a right neighbourhood of [math] to ]0,δ2[ such that the limits
[TABLE]
exist finite in [0,δ2[ and [0,+∞[, respectively. Under this assumption, we obtain in Proposition 7.4 the first and second terms of the asymptotic approximation of ut,γ(t) as t>0 tends to zero. In particular, for γ0=0 we see that
[TABLE]
as t tends to zero. Here, ui with i∈{1,2} denotes the harmonic solution of the exterior Dirichlet problem in R2∖Ωi with boundary datum fi and GΩo is the Green function of Ωo. We note that the limit value λ0 appears explicitly in the second asymptotic terms in the right hand side of (6). In Proposition 7.4 we also consider the case when γ0>0 and the holes shrink their size and mutual distance at a comparable speed. In such a case we compute the expansion
[TABLE]
as t tends to zero. Here u~ is the harmonic solution of a Dirichlet problem in the exterior domain R2∖Ω~(γ0) (see (46)) and HΩ~(γ0)2,1, HΩ~(γ0)1,2 are quantities related to the Green function in the exterior domain R2∖Ω~(γ0) (cf. Proposition 5.4).
To conclude this introduction, we observe that our result justifies the introduction of specific relations between the size and the distance when dealing with the Dirichlet problem in a domain with moderately close small holes. Conditions of this type appear also in other papers on the topic. For example, in [4], Bonnaillie-Noël, Dambrine, and Lacave have considered a Poisson problem with Dirichlet conditions in a domain with two moderately close holes. To compute the asymptotic expansion of the solution, they have assumed that the distance behaves like the size to some power β∈]0,1[. A condition which corresponds, with our notation, to the case when γ(t)=t(1−β)/β and the quotient (5) is constant and equal to 1−β. Another example can be found in [23], where Maz’ya and Movchan have analysed a Poisson problem with Dirichlet conditions in a domain with a large number small close holes. In such paper, it is assumed that the size is smaller than the distance to the power 7/4 (with our notation, γ(t)<t3/4) in order to obtain uniform approximations of the solution and that the size is smaller than the square of the distance (with our notation, γ(t)<t) to have approximations in H1 norm (see also Maz’ya, Movchan, and Nieves [26]).
The present paper is organised as follows. In Section 2 we present some preliminary results on the solution of the Dirichlet problem in a planar domain with many holes via potential theory. In Sections 3 and 4 we study some auxiliary integral operators that we use to convert problem (3) into integral equations, while in Section 5 we introduce some functions playing an important role in the description of the limiting behaviour of the solution uϵ1,ϵ2. In Section 6, we prove Theorem 6.6 on the representation of uϵ1,ϵ2 in terms of real analytic maps and known functions. In Section 7, we prove Proposition 7.4 where we analyse the asymptotic behaviour of the solution.
2 The Dirichlet problem in a domain with many holes
In this section, we present some results of classical potential theory and we show how to exploit them in order to solve the Dirichlet problem for the Laplace equation in a domain with many holes. The construction of the solution that we present here will be then used to convert problem (3) into equivalent integral equations. We start by denoting by S the function from R2∖{0} to R defined by
[TABLE]
As is well known, S is a fundamental solution for the Laplace operator in R2.
Let O be an open bounded subset of R2 of class C1,α. Let ϕ∈C0,α(∂O). Then vO[ϕ] denotes the single layer potential with density ϕ. Namely,
[TABLE]
where dσ denotes the arc length element on ∂O. As is well known, vO[ϕ] is a continuous function from R2 to R and the restrictions vO+[ϕ]≡vO[ϕ]∣clO and vO−[ϕ]≡vO[ϕ]∣Rn∖O belong to C1,α(clO) and to Cloc1,α(R2∖O), respectively. Here Cloc1,α(R2∖O) denotes the space of functions on R2∖O whose restrictions to clB belong to C1,α(clB) for all open bounded subsets B of R2∖O.
If ψ∈C1,α(∂O), then wO[ψ] denotes the double layer potential with density ψ. Namely,
[TABLE]
where νO denotes the outer unit normal to ∂O and the symbol ‘⋅’ denotes the scalar product in R2. The restriction wO[ψ]∣O extends to a function wO+[ψ] of C1,α(clO) and the restriction wO[ψ]∣Rn∖clO extends to a function wO−[ψ] of Cloc1,α(R2∖O).
Let
[TABLE]
for all ψ∈C0,α(∂O), and
[TABLE]
for all ϕ∈C1,α(∂O). As is well known (cf. Schauder [32, 33]) WO is compact from C1,α(∂O) to itself and WO∗ is compact from C0,α(∂O) to itself. In addition WO and WO∗ are adjoint with respect to the duality on C1,α(∂O)×C0,α(∂O) induced by L2(∂Ω) (cf. Kress [16]). As a consequence, one immediately deduces the validity of the following.
Lemma 2.1**.**
The operators ±21IO+WO are Fredholm of index [math] from C1,α(∂O) to itself. The operators ±21IO+WO∗ are Fredholm of index [math] from C0,α(∂O) to itself. The operator 21IO+WO∗ is the adjoint of 21IO+WO and the operator −21IO+WO∗ is the adjoint of −21IO+WO with respect to the duality on C1,α(∂O)×C0,α(∂O) induced by L2(∂Ω).
By exploiting the operators WO and WO∗ we can write the jump formulas
[TABLE]
which hold for all functions ψ∈C1,α(∂O) and ϕ∈C0,α(∂O) (cf., e.g., Folland [15, Chap. 3]). If ψ∈C1,α(∂O) then we also have
[TABLE]
Now assume that O has N connected components and R2∖clO has K+1 connected components and denote by O1, …, ON the (bounded) connected components of O and by O0−, O1−, …, OK− the connected components of R2∖clO. Since R2∖clO has a unique unbounded connected component we can assume that O1−, …, OK− are bounded and that O0− is unbounded.
In the sequel we exploit the following notation: if X is a subspace of L1(∂O) then we denote by X0 the subspace of X consisting of the functions which have zero integral mean.
Then we have the following classical lemma, where we describe the kernels of the integrals operator involved in the jump formulas in (7) (cf., e.g., Folland [15, Chap. 3]).
Lemma 2.2**.**
The following statements hold.
- (i)
The map from Ker(21IO+WO∗) to Ker(21IO+WO) which takes μ to v[∂O,μ]∣∂O is bijective.
2. (ii)
The map from Ker(−21IO+WO∗)0 to Ker(−21IO+WO) which takes μ to v[∂O,μ]∣∂O is one to one.
3. (iii)
Ker(21IO+WO)* consists of the functions from ∂O to R which are constant on ∂Oj− for all j∈{1,…,K} and which are identically equal to [math] on ∂O0−.*
4. (iv)
Ker(−21IO+WO)* consists of the functions from ∂O to R which are constant on ∂Oj, for all j∈{1,…,N}.*
5. (v)
If ϕ∈Ker(21IO+WO∗) and ∫∂Oj−ϕdσ=0 for all j∈{1,…,K}, then ϕ=0.
6. (vi)
If ϕ∈Ker(−21IO+WO∗) and ∫∂Ojϕdσ=0 for all j∈{1,…,N}, then ϕ=0.
7. (vii)
If ϕ∈Ker(−21IO+WO∗)0 and v[∂O,ϕ]∣∂O is constant on ∂O, then ϕ=0.
Moreover, by Lemma 2.2 (i), (iii), and (v) we deduce the validity of the following.
Lemma 2.3**.**
For each i∈{1,…,K} there exists a unique function τi∈C0,α(∂O) such that
[TABLE]
The set {τ1,…,τK} is a basis for Ker(21IO+WO∗) and the set {vO[τ1]∣∂O,…,vO[τK]∣∂O} is a basis for Ker(21IO+WO).
In the sequel we denote by XO,i the function from ∂O to R defined by
[TABLE]
where δi,j is the Kronecker delta function. By Lemma 2.2 (iii) it follows that {XO,1,…,XO,K} is a basis for Ker(21IO+WO). We also adopt the following notation, if Γ is a one dimensional manifold in R2, then its one dimensional Lebesgue measure is denoted by
∣Γ∣. Then we deduce the validity of the following.
Lemma 2.4**.**
Let ΛO≡(λOi,j)(i,j)∈{1,…,K}2 be the real K×K-matrix with entries λOi,j defined by
[TABLE]
Then ΛO is invertible and we have
vO[τi]∣∂O=∑j=1KλOi,jXO,j for all i∈{1,…,K}.
We are now ready to deduce the validity of the following Proposition 2.5, where we show how to construct the solution of the Dirichlet problem in a multiply perforated domain by solving some suitable integral equations.
Proposition 2.5**.**
Let g∈C1,α(∂O). Let u∈C1,α(clO) be the unique function such that Δu=0 and u∣∂O=g. Then the following statements hold:
- (i)
There exists and is unique a function μ∈C1,α(∂O) such that
[TABLE]
2. (ii)
We have
[TABLE]
Proof.
(i) By Lemma 2.3 one verifies that the right hand side of the first equation in (10) is orthogonal to
Ker(21IO+WO∗). Then the validity of the statement follows by Lemma (2.1) and by the standard properties of Fredholm operators.
(ii) It is a consequence of statement (i), of Lemma 2.4, of (7), of the mapping properties of single and double layer potentials, and of the uniqueness of the solution of the Dirichlet problem. ∎
3 The auxiliary maps M1 and M2
Proposition 2.5 shows how to construct the solution of the Dirichlet problem in two steps: first one constructs a basis for the kernel of the adjoint integral operator as in Lemma 2.3, then one finds the solution of the system of integral equations of (10). We want to exploit this approach for solving problem (3). Therefore, in this section, we perform the first of the two steps described above. Moreover, since our problem is defined in a domain which depends on ϵ1 and ϵ2, the integral equations delivered by Lemma 2.3 and Proposition 2.5 will be defined on an (ϵ1,ϵ2)-dependent domain as well. As we are going to show, we will get rid of this dependence by performing a convenient change of variables.
We now introduce the auxiliary maps M1 and M2 representing the counterpart of Lemma 2.3. For all i∈{1,2} we denote by Mi≡(Mio,Mi,1,Mi,2,Mic) the map from ]−δ1,δ1[×]−δ2,δ2[×C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) to C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2)×R2 defined by
[TABLE]
for all (ϵ1,ϵ2,ρio,ρi,1,ρi,2)∈]−δ1,δ1[×]−δ2,δ2[×C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2).
Then, by a straightforward computation based on the rule of change of variable in integrals and by Lemma 2.3, one deduces the validity of the following Proposition 3.1.
Proposition 3.1**.**
Let i∈{1,2}. If ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}, then we have
[TABLE]
if and only if
[TABLE]
with τi∈C0,α(∂Ω(ϵ1,ϵ2)) defined by
[TABLE]
Moreover, there exists a unique triple (ρio[ϵ1,ϵ2],ρi,1[ϵ1,ϵ2],ρi,2[ϵ1,ϵ2])∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) such that Mi[ϵ1,ϵ2,ρio[ϵ1,ϵ2],ρi,1[ϵ1,ϵ2],ρi,2[ϵ1,ϵ2]]=0.
We now pass to consider the case when ϵ2=0 in Proposition 3.2 and the case when ϵ1=0 and ϵ2=0 in Proposition 3.3. The proofs of Propositions 3.2 and 3.3 can be effected by straightforward computations and by exploiting Lemma 2.2.
Proposition 3.2**.**
Let i∈{1,2}. If ϵ1∈]−δ1,δ1[ (and ϵ2=0), then we have
[TABLE]
if and only if
[TABLE]
Moreover, there exists a unique triple (ρio[ϵ1,0],ρi,1[ϵ1,0],ρi,2[ϵ1,0])∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) such that Mi[ϵ1,0,ρio[ϵ1,0],ρi,1[ϵ1,0],ρi,2[ϵ1,0]]=0.
We also observe that Proposition 3.2 implies that
[TABLE]
(see also Lemma 2.2 (vi)). In the following Proposition 3.3 we exploit the definition of Ω~(ϵ2) introduced in (2) and we consider the case ϵ1=0.
Proposition 3.3**.**
Let i∈{1,2}. If ϵ2∈]−δ2,δ2[∖{0} (and ϵ1=0), then we have
[TABLE]
if and only if
[TABLE]
with ρ~i∈C0,α(∂Ω~(ϵ2)) defined by
[TABLE]
Moreover, there exists a unique triple (ρio[0,ϵ2],ρi,1[0,ϵ2],ρi,2[0,ϵ2])∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) such that Mi[0,ϵ2,ρio[0,ϵ2],ρi,1[0,ϵ2],ρi,2[0,ϵ2]]=0.
Our aim is now to show that (ρio[ϵ1,ϵ2],ρi,1[ϵ1,ϵ2],ρi,2[ϵ1,ϵ2]) depends analytically on (ϵ1,ϵ2). In order to do so, we plan to apply the implicit function theorem for real analytic maps in Banach space. Thus, we need to show the real analyticity of Mi and the invertibility of the partial differential of Mi. We do that in the following technical Lemma 3.4.
Lemma 3.4**.**
Let i∈{1,2}. The following statements hold.
- (i)
The map Mi is real analytic from ]−δ1,δ1[×]−δ2,δ2[×C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) to C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2)×R2.
2. (ii)
Let (ϵˉ1,ϵˉ2,ρˉio,ρˉi,1,ρˉi,2)∈]−δ1,δ1[×]−δ2,δ2[×C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2), then
[TABLE]
(the partial differential of Mi with respect to (ρio,ρi,1,ρi,2) evaluated at (ϵˉ1,ϵˉ2,ρˉio,ρˉi,1,ρˉi,2)) is an isomorphism from C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) to C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2)×R2.
Proof.
The validity of statement (i) follows by standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]) and by classical mapping properties of layer potentials (cf., e.g., Miranda [30]).
To prove statement (ii) we observe that the partial differential (14) is delivered by
[TABLE]
for all (ρio,ρi,1,ρi,2)∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2). By classical potential theory (cf. Section 2) and by a standard argument based on the theorem of change of variables in integrals one verifies that for all fixed (go,g1,g2,c1,c2)∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2)×R2 there exists and is unique a triple (ρio,ρi,1,ρi,2)∈C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) such that
[TABLE]
Then the validity of statement (ii) follows by the open mapping theorem.∎
Then, by a standard argument based on the implicit function theorem for real analytic maps (cf. Deimling [14]) we deduce the following Proposition 3.5.
Proposition 3.5**.**
Let i∈{1,2}. Then the map from ]−δ1,δ1[×]−δ2,δ2[ to C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) which takes (ϵ1,ϵ2) to (ρio[ϵ1,ϵ2],ρi,1[ϵ1,ϵ2],ρi,2[ϵ1,ϵ2]) is real analytic. Moreover, the set of zeros of Mi in ]−δ1,δ1[×]−δ2,δ2[×C0,α(∂Ωo)×C0,α(∂Ω1)×C0,α(∂Ω2) coincides with the graph of (ρio[⋅,⋅],ρi,1[⋅,⋅],ρi,2[⋅,⋅]).
4 The auxiliary map L
As we have done in the previous section for the counterpart of Lemma 2.3 for our problem (3), we now turn to consider the corresponding statement for the system in (10) of Proposition 2.5. Also in this case, we find convenient to perform a change of variables and to introduce the auxiliary map L≡(Lo,L1,L2) from ]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 to C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2) defined by
[TABLE]
for all (ϵ1,ϵ2,θo,θ1,θ2)∈]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0.
Then, by a straightforward computation based on the rule of change of variable in integrals and by Proposition 2.5, one deduces the validity of the following Proposition 4.1.
Proposition 4.1**.**
If ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}, then we have
[TABLE]
if and only if
[TABLE]
with ϕ,f∈C1,α(∂Ω(ϵ1,ϵ2)) defined by
[TABLE]
and XΩ(ϵ1,ϵ2),k, τk defined as in (9) and (11), respectively.
Moreover, there exists a unique triple (θo[ϵ1,ϵ2],θ1[ϵ1,ϵ2],θ2[ϵ1,ϵ2])∈C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 such that L[ϵ1,ϵ2,θo[ϵ1,ϵ2],θ1[ϵ1,ϵ2],θ2[ϵ1,ϵ2]]=0.
We now pass to consider the case when ϵ2=0 in Proposition 4.2 and the case when ϵ1=0 and ϵ2=0 in Proposition 4.3.
Proposition 4.2**.**
If ϵ1∈]−δ1,δ1[ (and ϵ2=0), then we have
[TABLE]
if and only if
[TABLE]
Moreover, there exists a unique triple (θo[ϵ1,0],θ1[ϵ1,0],θ2[ϵ1,0])∈C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 such that L[ϵ1,0,θo[ϵ1,0],θ1[ϵ1,0],θ2[ϵ1,0]]=0.
Proof.
If θo satisfies the first equation of the system (15), then by the properties of adjoint operators, by Proposition 3.2, and by the definition of the double layer potential we have
[TABLE]
Then the validity of the proposition follows by a straightforward computation based on the rule of change of variable in integrals, by equality (12), and by a standard argument based on Lemma 2.2 and Proposition 3.2.
∎
Then we turn to consider the case ϵ1=0.
Proposition 4.3**.**
If ϵ2∈]−δ2,δ2[ (and ϵ1=0), then we have
[TABLE]
if and only if
[TABLE]
with θ~,f~∈C1,α(∂Ω~(ϵ2)) defined by
[TABLE]
and XΩ~(ϵ2),h, ρ~h defined as in (9) and (13), respectively.
Moreover, there exists a unique triple (θo[0,ϵ2],θ1[0,ϵ2],θ2[0,ϵ2])∈C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 such that L[0,ϵ2,θo[0,ϵ2],θ1[0,ϵ2],θ2[0,ϵ2]]=0.
Proof.
If θo satisfies the first equation of the system (16), then by the properties of adjoint operators, by Proposition 3.3, and by the definition of the double layer potential we have
[TABLE]
Then the validity of the proposition follows by a straightforward computation based on the rule of change of variable in integrals and by a standard argument based on Lemma 2.2 and Proposition 3.3.
∎
In the following Proposition 4.4 we show an orthogonality property of the operator L.
Proposition 4.4**.**
We have
[TABLE]
for all h∈{1,2} and for all (ϵ1,ϵ2,θo,θ1,θ2)∈]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0.
Proof.
Let ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}. Let ϕ,f∈C1,α(∂Ω(ϵ1,ϵ2)) be defined as in Proposition 4.1. Then the validity of (17) follows by equality
[TABLE]
by the orthogonality of Ran(21IΩ(ϵ1,ϵ2)+WΩ(ϵ1,ϵ2)) and of Ker(21IΩ(ϵ1,ϵ2)+WΩ(ϵ1,ϵ2)∗), by Proposition 3.1, and by a straightforward computation.
If at least one of ϵ1 and ϵ2 is [math], then, by Propositions 4.2 and 4.3, by the properties of adjoint operators, and by the definition of the double layer potential, we have
[TABLE]
If ϵ2=0, then the orthogonality of Ran(−21IΩh+WΩh) and of Ker(−21IΩh+WΩh∗) and equality
[TABLE]
(cf. Proposition 3.2) imply that
[TABLE]
If instead ϵ1=0 and ϵ2=0, then by the orthogonality of Ran(−21IΩ~(ϵ2)+WΩ~(ϵ2)) and of Ker(−21IΩ~(ϵ2)+WΩ~(ϵ2)∗) and equality
[TABLE]
where ρ~h∈C0,α(∂Ω~(ϵ2)) is defined as in Proposition 3.3, we deduce that
[TABLE]
Now the validity of (17) for ϵ1=0 or ϵ2=0 follows by (18), (19), (20), and by a straightforward computation.
∎
As done in Section 3, we plan to apply (a corollary of) the implicit function theorem to prove the real analyticity of (θo[⋅,⋅],θ1[⋅,⋅],θ2[⋅,⋅]). In order to do so, in the following technical Lemma 4.5, we study the regularity of L and its partial differential.
Lemma 4.5**.**
The following statements hold.
- (i)
The map L is real analytic from ]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 to C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2).
2. (ii)
Let (ϵˉ1,ϵˉ2,θˉo,θˉ1,θˉ2)∈]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0, then
[TABLE]
(the partial differential of L with respect to the variable (θo,θ1,θ2) evaluated at (ϵˉ1,ϵˉ2,θˉo,θˉ1,θˉ2)) is an isomorphism from C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 onto the subspace of C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2) consisting of those triples (ψo,ψ1,ψ2) such that
[TABLE]
Proof.
Statement (i) follows by the standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]) and by classical mapping properties of layer potentials (cf., e.g., Miranda [30]).
To prove statement (ii) we observe that the partial differential (21) is delivered by
[TABLE]
for all (θo,θ1,θ2)∈C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0. Then we take a triple (ψo,ψ1,ψ2) in C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2) which satisfies condition (22) and, by arguing as in the proof of Propositions 4.1, 4.2, 4.3, we verify that there exist a unique triple (θo,θ1,θ2)∈C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 such that
[TABLE]
Now the validity of the statement (ii) follows by the open mapping theorem and by Proposition 4.4.
∎
We now introduce in the following Lemma 4.6 a technical corollary of the implicit function theorem for real analytic maps. For a proof we refer to Lanza de Cristoforis [18, Thm. 13].
Lemma 4.6**.**
Let X, Y, Z, Z1 be Banach spaces. Let O be an open subset of X×Y such that (xˉ,yˉ)∈O. Let F be a real analytic map from O to Z such that F(xˉ,yˉ)=0. Let the partial differential ∂yF(xˉ,yˉ) with respect to the variable y be an homeomorphism from Y onto its image V≡Ran(∂yF(xˉ,yˉ)). Assume that there exists a closed subspace V1 of Z such that Z=V⊕V1. Let O1 be an open subset of X×Y×Z containing (xˉ,yˉ,0) and such that (x,y,F(x,y)) and (x,y,0) belong to O1 for all (x,y)∈O. Let G be a real analytic map from O1 to Z1 such that G(x,y,F(x,y))=0 for all (x,y)∈O, G(x,y,0)=0 for all (x,y)∈O, and such that the partial differential ∂zG(xˉ,yˉ,0) is surjective onto Z1 and has kernel equal to V. Then there exist an open neighbourhood U of xˉ in X, an open neighbourhood V of yˉ in Y with U×V⊆O, and a real analytic map T from U to V such that the set of zeros of F in U×V coincides with the graph of T.
We are finally in the position to apply Lemma 4.6 to equation L[ϵ1,ϵ2,θo,θ1,θ2]=0 and prove that the triple (θo[ϵ1,ϵ2],θ1[ϵ1,ϵ2],θ2[ϵ1,ϵ2]) depends analytically on (ϵ1,ϵ2).
Proposition 4.7**.**
The function from ]−δ1,δ1[×]−δ2,δ2[ to C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 which takes (ϵ1,ϵ2) to (θo[ϵ1,ϵ2],θ1[ϵ1,ϵ2],θ2[ϵ1,ϵ2]) is real analytic. Moreover, the set of zeros of L in ]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 coincides with the graph of (θo[⋅,⋅],θ1[⋅,⋅],θ2[⋅,⋅]).
Proof.
Let (ϵˉ1,ϵˉ2,θˉo,θˉ1,θˉ2)∈]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0 be such that L[ϵˉ1,ϵˉ2,θˉo,θˉ1,θˉ2]=0. Let X≡R2, Y≡C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0, Z≡C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2), Z1≡R2, O≡]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0. Let F≡L. Let xˉ≡(ϵˉ1,ϵˉ2) and yˉ≡(θˉo,θˉ1,θˉ2). Let V be the subspace of C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2) consisting of the triples (ψo,ψ1,ψ2) which satisfy the condition in (22) with ϵ1=ϵˉ1 and ϵ2=ϵˉ2, let V1 be the 2-dimensional subspace of C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2) generated by (ρ1o[ϵˉ1,ϵˉ2],ρ1,1[ϵˉ1,ϵˉ2],ρ1,2[ϵˉ1,ϵˉ2]) and (ρ2o[ϵˉ1,ϵˉ2],ρ2,1[ϵˉ1,ϵˉ2],ρ2,2[ϵˉ1,ϵˉ2]). Let
O1≡]−δ1,δ1[×]−δ2,δ2[×C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0×C1,α(∂Ωo)×C1,α(∂Ω1)×C1,α(∂Ω2).
Let G≡(G1,G2) be defined by
[TABLE]
for all h∈{1,2} and for all (ϵ1,ϵ2,θo,θ1,θ2,ψo,ψ1,ψ2)∈O1. Then Lemma 4.6 implies that there exist an open neighbourhood of U of (ϵˉ1,ϵˉ2) in ]−δ1,δ1[×]−δ2,δ2[, an open neighbourhood V of (θˉo,θˉ1,θˉ2) in C1,α(∂Ωo)×C1,α(∂Ω1)0×C1,α(∂Ω2)0, and a real analytic map T≡(To,T1,T2) from U to V such that the set of zeros of L in U×V coincides with the graph of T. Then Propositions 4.1, 4.2, and 4.3 imply that T[ϵ1,ϵ2]=(θo[ϵ1,ϵ2],θ1[ϵ1,ϵ2],θ2[ϵ1,ϵ2]) for all (ϵ1,ϵ2)∈U and the validity of the proposition follows.
∎
5 The auxiliary functions HxΩo, HΩ1x, HΩ2x, and HΩ~(ϵ2)x
In the next Section 6, we will exploit the results of Sections 3 and 4 and the representation formula of Proposition 2.5 to describe the dependence of the solution uϵ1,ϵ2 of (3) in terms of analytic functions of ϵ1, ϵ2 and of elementary functions of log∣ϵ1∣ and log∣ϵ1ϵ2∣. Before doing so, we introduce in this section the auxiliary functions HxΩo, HΩ1x, HΩ2x, and HΩ~(ϵ2)x, which will play an important role in the description of the limit behaviour of uϵ1,ϵ2. We note that HxΩo(y) is the difference between the Dirichlet Green function in Ωo and the fundamental solution S(x−y) (see (64)). Analogous relations hold for HΩ1x(y), HΩ2x(y), and HΩ~(ϵ2)x(y) in the exterior domains R2∖Ω1, R2∖Ω2, and R2∖Ω~(ϵ2), respectively.
Proposition 5.1**.**
Let x∈Ωo be fixed. Let HxΩo∈C1,α(clΩo) be the solution of
[TABLE]
Then vΩo[ρjo[ϵ1,0]](x)=−HxΩo(ϵ1pj) and vΩo[ρjo[0,ϵ2]](x)=−HxΩo(0) for all (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[ and for all j∈{1,2}.
Proof.
Let u∈C1,α(clΩo) and Δu=0 in Ωo. Then by classical potential theory there exists μ∈C1,α(∂Ωo) such that u=wΩo+[μ] (cf. Section 2). Then, by the jump properties of the double layer potential (see (7)), by standard properties of adjoint operators, and by Proposition 3.2, we have
[TABLE]
It follows that
[TABLE]
The proof of vΩo[ρjo[0,ϵ2]](x)=−HxΩo(0) is similar. Indeed, for u and μ as above we have
[TABLE]
(see also Proposition 3.3) and thus
[TABLE]
∎
Proposition 5.2**.**
Let h∈{1,2} and x∈R2∖∂Ωh be fixed. Let HΩhx∈Cloc1,α(R2∖Ωh) be the solution of
[TABLE]
Then
[TABLE]
If in addition x∈clΩh, then we have
[TABLE]
In particular, limy→∞HΩhx(y)=limy→∞HΩh0(y) for all x∈clΩh and all h∈{1,2}.
Proof.
We first prove (25). Let u∈Cloc1,α(R2∖Ωh), Δu=0 in R2∖clΩh, and supy∈R2∖Ωh∣u(y)∣<+∞. Then, by classical potential theory there exists μ∈C1,α(∂Ωh) such that u=wΩh−[μ]+limy→∞u(y) (cf. Folland [15, Ch. 3], see also Section 2). Then by the jump properties of the double layer potential (7), by standard properties of adjoint operators, and by Proposition 3.2, we have
[TABLE]
Thus
[TABLE]
To prove (26) we observe that, by Proposition 3.2 and by the jump properties of the normal derivative of the single layer potential (cf. (7)), we have νΩh⋅∇vΩh+[ρj,h[ϵ1,0]]∣∂Ωh=0. We deduce that vΩh[ρj,h[ϵ1,0]] is constant on clΩh and the validity of statement (ii) follows.
∎
In the proof of Proposition 5.4 here below we exploit the following result of potential theory.
Lemma 5.3**.**
Let ϵ2∈]−δ2,δ2[∖{0} and let h,k∈{1,2} with h=k. Then the operator from C1,α(∂Ω~(ϵ2)) to itself which takes μ to the function defined by
[TABLE]
is a linear isomorphism.
Proof.
By Proposition 2.1 and by standard properties of Fredholm operators one verifies that the operator from C1,α(∂Ω~(ϵ2)) to itself which takes a function μ to the function defined by (28) is Fredholm of index [math]. Thus, in order to show that it is an isomorphism it suffices to show that μ=0 when
[TABLE]
If μ satisfies equation (29), then by the jump properties of the double layer potential (cf. (7)) we have
[TABLE]
for all x∈∂Ω~(ϵ2). We observe that both the left and the right hand side of (30) define functions which are bounded in R2∖Ω~(ϵ2). Accordingly, the uniqueness properties of the solution of the exterior Dirichlet problem (cf., e.g., Folland [15, Chap. 2]) implies that equality (30) holds for all x∈R2∖Ω~(ϵ2). Then, by the decay properties of wΩ~(ϵ2)−[μ](x) and of S(x−ph)−S(x−pk) as x→∞ we deduce that
[TABLE]
Now we observe that by equality (8) and by the divergence theorem we have
[TABLE]
Moreover, by the definition of the double layer potential and by equalities wΩh(1,ϵ2)[1](pk)=0 and wΩh(1,ϵ2)[1](ph)=1 (cf. Section 2, see also Folland [15, Chap. 3]) we have
[TABLE]
Hence, by equalities (30), (32), and (33) we deduce that
[TABLE]
Then, by equalities (30), (31), and (34), and by Lemma 2.2 it follows that μ=0. Our proof is now completed.
∎
We observe here that Lemma 5.3 implies that
[TABLE]
for all u∈Cloc1,α(R2∖Ω~(ϵ2)) such that Δu=0 in R2∖clΩ~(ϵ2) and supy∈R2∖Ω~(ϵ2)∣u(y)∣<+∞ (see also Folland [15, Chap. 2]).
Proposition 5.4**.**
Let ϵ2∈]−δ2,δ2[∖{0} be fixed. For each x∈R2∖∂Ω~(ϵ2) let HΩ~(ϵ2)x∈Cloc1,α(R2∖Ω~(ϵ2)) be the solution of
[TABLE]
Let j∈{1,2}. Let ρ~j∈C0,α(∂Ω~(ϵ2)) be defined as in (13). Let HΩ~(ϵ2)j,i∈R be defined by
[TABLE]
Let h,k∈{1,2} with h=k. Then
[TABLE]
for all ξ∈R2 such that ph+ϵ2ξ∈/∂Ω~(ϵ2).
In addition, if ξ∈clΩh, then
[TABLE]
and if ξ∈(pk−ph)/ϵ2+clΩk, then
[TABLE]
Proof.
Let u∈Cloc1,α(R2∖Ω~(ϵ2)), Δu=0 in R2∖clΩ~(ϵ2), and supy∈R2∖Ω~(ϵ2)∣u(y)∣<+∞. Then, by classical potential theory there exists μ∈C1,α(∂Ω~(ϵ2)) such that
[TABLE]
(cf. Lemma 5.3). Then, a computation based on the divergence theorem, on equality (8), and on equality wΩk(1,ϵ2)[1](pk)=1, shows that
[TABLE]
Hence, by the jump properties of the double layer potential (7) and by the decay at ∞ of wΩh−[μ](x) and S(x−pk)−S(x−ph) we obtain that
[TABLE]
Now let ρ~j∈C0,α(∂Ω~(ϵ2)) be defined as in (13). Then by the jump properties of the double layer potential (7), by the definition of the single layer potential (cf. Section 2), by standard properties of adjoint operators, and by Proposition 3.3, we have
[TABLE]
Then, by the rule of change of variables in integrals and by (36) we deduce that
[TABLE]
for all ξ∈R2 such that ph+ϵ2ξ∈/∂Ω~(ϵ2). It follows that the first equality in (38) holds with HΩ~(ϵ2)j,k and HΩ~(ϵ2)j,h as in (37). Then, by (35) one deduces the validity of the second equality in (38). To prove (39) and (40) we observe that, by Proposition 3.2 and by the jump properties of the single layer potential (7), we have νΩ~(ϵ2)⋅∇vΩ~(ϵ2)+[ρ~j]∣∂Ω~(ϵ2)=0. Thus vΩ~(ϵ2)+[ρ~j] is constant in clΩh(1,ϵ2) and in clΩk(1,ϵ2) and the validity of (39) and (40) follows by (37) and by a straightforward computation based on the rule of change of variables in integrals.
∎
6 Representation of uϵ1,ϵ2 in terms of analytic maps
In this section, we prove our main Theorem 6.6 on the representation of uϵ1,ϵ2 in terms of real analytic maps and known functions. We will do so by exploiting the representation formula of Proposition 2.5, the real analyticity results of Propositions 3.5 and 4.7, and the auxiliary functions of Sections 5.
In the following Propositions 6.1–6.5 we introduce the functions U[ϵ1,ϵ2] and V[ϵ1,ϵ2], the vector F[ϵ1,ϵ2], and the matrices R[ϵ1,ϵ2] and Λ(ϵ1,ϵ2) which we exploit to write uϵ1,ϵ2 and uϵ1,ϵ2(ϵ1p1+ϵ1ϵ2⋅) in terms of real analytic maps (cf. Theorem 6.6).
Proposition 6.1**.**
For each (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[ there exists a unique function U[ϵ1,ϵ2] in C1,α(clΩ(ϵ1,ϵ2)) such that
[TABLE]
for all x∈clΩo∖(clΩ1(ϵ1,ϵ2)∪clΩ2(ϵ1,ϵ2)). Moreover, the following statements hold.
- (i)
Let ΩM be an open subset of Ωo such that 0∈/clΩM. Let δM∈]0,δ1] be such that clΩM∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[ and for all k∈{1,2}. Then there exists a real analytic map UM from ]−δM,δM[×]−δ2,δ2[ to C1,α(clΩM) such that
[TABLE]
where uo∈C1,α(clΩo) is the unique solution of
[TABLE]
2. (ii)
Let h,k∈{1,2} and h=k. Let Ωm be an open bounded subset of R2∖clΩh. Let δm∈]0,δ1] be such that ϵ1ph+ϵ1ϵ2clΩm⊆Ωo and (ϵ1ph+ϵ1ϵ2clΩm)∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δm,δm[2. Then there exists a real analytic map Uhm from ]−δm,δm[2 to C1,α(clΩm) such that
[TABLE]
Moreover,
[TABLE]
where uh∈Cloc1,α(R2∖Ωh) is the solution of
[TABLE]
and
[TABLE]
for all ξ∈clΩm and all ϵ2∈]−δm,δm[∖{0},
where u~∈Cloc1,α(R2∖Ω~(ϵ2)) is the solution of
[TABLE]
with f~(x)≡fj((x−pj)/ϵ2) for all j∈{1,2} and x∈∂Ωj(1,ϵ2), and where w~∈Cloc1,α(R2∖Ω~(ϵ2)) is the solution of
[TABLE]
Proof.
We first consider statement (i). We observe that by Propositions 4.2 and 4.3 we have θo[0,ϵ2]=θo[ϵ1,0]=μo for all (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[, where μo∈C1,α(∂Ωo) is the unique solution of (21IΩo+WΩo)μo=fo. By standard properties of real analytic maps it follows that there is a real analytic map Θo from ]−δ1,δ1[×]−δ2,δ2[ to C1,α(∂Ωo) such that θo[ϵ1,ϵ2]=μo+ϵ1ϵ2Θo[ϵ1,ϵ2] for all (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[. Since wΩo+[μo]=uo by the jump formula (7) and by the uniqueness of the solution of the Dirichlet problem, we deduce that
[TABLE]
Then we define
[TABLE]
for all x∈clΩM and for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[. One readily verifies the validity of (42). In addition, by the standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]), by the classical mapping properties of layer potentials (cf., e.g., Miranda [30]), and by Proposition 4.7 one verifies that the map from ]−δM,δM[×]−1,1[ to C1,α(clΩM) which takes (ϵ1,ϵ2) to UM[ϵ1,ϵ2] is real analytic.
We now prove statement (ii). We define
[TABLE]
for all (ϵ1,ϵ2)∈]−δm,δm[2. Then, by the standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]) and by the classical mapping properties of layer potentials (cf., e.g., Miranda [30]) we verify that the map which takes (ϵ1,ϵ2) to Uhm[ϵ1,ϵ2] is real analytic from ]−δm,δm[2 to C1,α(clΩm). The validity of equality (43) can be deduced by a straightforward computation based on the rule of change of variables in integrals. We now verify (44). A straightforward computation shows that
[TABLE]
Then we observe that by Proposition 4.2 and by the jump formulae (7) we have
[TABLE]
In addition, by Proposition 4.2 and by the jump formulae (7), we have
[TABLE]
Accordingly, equality (27) implies that
[TABLE]
and by the uniqueness of the solution of the exterior Dirichlet problem we deduce that
[TABLE]
Now, equality (44) follows by (47), (48), and (49). The proof of (45) is similar. By a straightforward computation based on the rule of change of variables in integrals we verify that
[TABLE]
where θ~∈C1,α(∂Ω~(ϵ2)) is defined by
[TABLE]
By Proposition 4.3, we have
[TABLE]
By Proposition 4.3, by the jump formulae (7), by equality (41), and by definition (37), we have
[TABLE]
Then, by the uniqueness of the solution of the exterior Dirichlet problem, we deduce that
[TABLE]
Hence, the validity of (45) follows by (50), (51), and (52).
∎
Proposition 6.2**.**
For all (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[ we denote by V[ϵ1,ϵ2]≡(V1[ϵ1,ϵ2],V2[ϵ1,ϵ2]) the function of C1,α(clΩ(ϵ1,ϵ2))2 defined by
[TABLE]
for all j∈{1,2}.
Then the following statements hold.
- (i)
Let ΩM be an open subset of Ωo such that 0∈/clΩM. Let δM∈]0,δ1] be such that clΩM∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[ and for all k∈{1,2}. Then there exists a real analytic map VM≡(V1M,V2M) from ]−δM,δM[×]−δ2,δ2[ to C1,α(clΩM)2 such that
[TABLE]
Moreover,
[TABLE]
and
[TABLE]
2. (ii)
Let h,k∈{1,2} and h=k. Let Ωm be an open bounded subset of R2∖clΩh. Let δm∈]0,δ1] be such that ϵ1ph+ϵ1ϵ2clΩm⊆Ωo and (ϵ1ph+ϵ1ϵ2clΩm)∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δm,δm[2. Then there exists a real analytic map Vhm≡(Vh,1m,Vh,2m) from ]−δm,δm[2 to C1,α(clΩm) such that
[TABLE]
for all j∈{1,2}, (ϵ1,ϵ2)∈(]−δm,δm[∖{0})2. Moreover,
[TABLE]
for all j∈{1,2}, ξ∈clΩm, and ϵ1∈]−δm,δm[, and
[TABLE]
for all j∈{1,2}, ξ∈clΩm, and ϵ2∈]−δm,δm[∖{0}.
Proof.
To prove statement (i) we take
[TABLE]
Then, the real analyticity of VM follows by the standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]), by the classical mapping properties of layer potentials (cf., e.g., Miranda [30]), and by Proposition 3.5. The validity of equalities (53) and (54) can be deduced by Proposition 5.1 and by Propositions 3.2 and 3.3.
We now consider statement (ii). We define
[TABLE]
for all j∈{1,2} and (ϵ1,ϵ2)∈]−δm,δm[2. Then, by the standard properties of integral operators with real analytic kernels and with no singularity (see, e.g., Lanza de Cristoforis and the second author [21]), by the mapping properties of layer potentials (cf., e.g., Miranda [30]), and by Proposition 3.5 we verify that Vhm≡(Vh,1m,Vh,2m) is real analytic. Then equality (55) follows by a straightforward computation based on the rule of change of variables in integrals and on Proposition 3.1.
To prove equality (56) we observe that by Proposition 3.2
[TABLE]
Then the validity of (56) follows by Proposition 5.1 and equality (25). By Proposition 5.1 and by equality (38) one verifies (57).
∎
Proposition 6.3**.**
Let F≡(F1,F2) be the function from ]−δ1,δ1[×]−δ2,δ2[ to R2 defined by
[TABLE]
Then F is real analytic. Moreover, we have
[TABLE]
for all j,h,k∈{1,2}, h=k, ϵ1∈]−δ1,δ1[, and ϵ2∈]−δ2,δ2[∖{0}.
Proof.
The real analyticity of F is a consequence of Proposition 3.5. The validity of (58) follows by (23) and (27). To prove (59) one observes that
[TABLE]
with f~(x)≡fh((x−ph)/ϵ2) for all h∈{1,2} and x∈∂Ωh(1,ϵ2) and ρ~j as in Proposition 3.3. Then the validity of (59) follows by (24), (37), and (41).
∎
Here below M2×2(R) denotes the space of the 2×2 real matrices.
Proposition 6.4**.**
Let R≡(Ri,j)(i,j)∈{1,2}2 be the function from ]−δ1,δ1[×]−δ2,δ2[ to M2×2(R) defined by
[TABLE]
for all (ϵ1,ϵ2)∈]−δ1,δ1[×]−δ2,δ2[ and for all i,j,k∈{1,2} with j=k. Then R is real analytic and
[TABLE]
for all i,j∈{1,2}, ϵ1∈]−δ1,δ1[, and ϵ2∈]−δ2,δ2[∖{0}.
Proof.
The real analyticity of R is a consequence of Proposition 3.5 and of the mapping properties of the single layer potential. Equality (60) follows by Proposition 5.1, by (25), and by Proposition 3.1. Equality (61) follows by Proposition 5.1 and by equality (39).
∎
Proposition 6.5**.**
Let ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}. Then the matrix \Lambda(\epsilon_{1},\epsilon_{2})\equiv\bigl{(}\Lambda_{i,j}(\epsilon_{1},\epsilon_{2})\bigr{)}_{(i,j)\in\{1,2\}^{2}} defined by
[TABLE]
satisfies the equality
[TABLE]
with τi∈C0,α(∂Ω(ϵ1,ϵ2)) defined as in (11). In particular, the matrix Λ(ϵ1,ϵ2) is invertible.
Proof.
Equality (62) follows by Proposition 3.1 and by the rule of change of variables in integrals. The invertibility of Λ(ϵ1,ϵ2) is a consequence of Lemma 2.4. ∎
We are now ready to prove our main Theorem 6.6, where we introduce representation formulas for uϵ1,ϵ2 and for uϵ1,ϵ2(ϵ1ph+ϵ1ϵ2⋅) in terms of real analytic functions of the pair (ϵ1,ϵ2) and of elementary functions of log∣ϵ1∣ and log∣ϵ1ϵ2∣. In the sequel, At denotes the transpose of a matrix A and A−1 denotes the inverse of an invertible matrix A.
Theorem 6.6**.**
The following statements hold.
- (i)
Let ΩM be an open subset of Ωo such that 0∈/clΩM. Let δM∈]0,δ1] be such that clΩM∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[ and for all k∈{1,2}. Then
[TABLE]
for all ϵ1∈]−δM,δM[∖{0} and ϵ2∈]−δ2,δ2[∖{0}.
2. (ii)
Let h,k∈{1,2} and h=k. Let Ωm be an open bounded subset of R2∖clΩh. Let δm∈]0,δ1] be such that ϵ1ph+ϵ1ϵ2clΩm⊆Ωo and (ϵ1ph+ϵ1ϵ2clΩm)∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δm,δm[2. Then
[TABLE]
for all (ϵ1,ϵ2)∈(]−δm,δm[∖{0})2, where Sh(ϵ1,ϵ2)∈R2 is defined by
[TABLE]
Proof.
By Propositions 2.5 we have
[TABLE]
with ϕ as in Proposition 4.1, τ1 and τ2 as in (11), and Λ(ϵ1,ϵ2) as in (62). Then the validity of (i) and (ii) follows by Propositions 6.1, 6.2, 6.3, and 6.5, and by a computation based on the rule of change of variables in integrals.
∎
7 Asymptotic behaviour of uϵ1,ϵ2 as (ϵ1,ϵ2)→(0,γ0)
In this section we show how Theorem 6.6 can be exploited to obtain asymptotic approximations of the solution of problem (3) as the pair of parameters (ϵ1,ϵ2) approaches a degenerate pair (0,γ0). As we shall see, the function 1/log∣ϵ1ϵ2∣ will appear in many of our expressions and, in order that such expressions make sense, we have to ensure that ∣ϵ1ϵ2∣<1 in the admissible set. Then, we shrink δ1 and we assume that in this section we have
[TABLE]
In the following Proposition 7.1 we describe the inverse matrix Λ(ϵ1,ϵ2)−1. In the sequel, A∗ denotes the cofactor matrix of a matrix A, so that A∗t is the adjugate of A.
Proposition 7.1**.**
Let ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}. Then we have
[TABLE]
and
[TABLE]
with
[TABLE]
We observe that, since Λ(ϵ1,ϵ2) is invertible by Proposition 6.5, we have that Rϵ1,ϵ2=0 for all ϵ1∈]−δ1,δ1[∖{0} and ϵ2∈]−δ2,δ2[∖{0}.
In the following Proposition 7.2 we write a convenient expression for uϵ1,ϵ2∣clΩM. We exploit the following definition
[TABLE]
(cf. Proposition 5.1). We observe that GΩo is the Dirichlet Green function for the domain Ωo.
Proposition 7.2**.**
Let ΩM be an open subset of Ωo such that 0∈/clΩM. Let δM∈]0,δ1] be such that clΩM∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[ and for all k∈{1,2}. Then there exists a real analytic map XM≡(X1M,X2M) from ]−δM,δM[×]−δ2,δ2[ to C1,α(clΩM)2 such that
[TABLE]
for all ϵ1∈]−δM,δM[∖{0} and ϵ2∈]−δ2,δ2[∖{0}.
Proof.
By Proposition 6.2 (i) and by standard properties of real analytic functions there exists a real analytic map XM≡(X1M,X2M) from ]−δM,δM[×]−δ2,δ2[ to C1,α(clΩM)2 such that
[TABLE]
Then, by a straightforward computation based on Proposition 7.1 we have
[TABLE]
for all ϵ1∈]−δM,δM[∖{0} and ϵ2∈]−δ2,δ2[∖{0}. Now the validity of the statement follows by Theorem 6.6.
∎
We now observe that if we try to pass to the limit in the representation formula (65) we face the problem that
[TABLE]
does not exist when γ0=0. As it has been announced in the introduction, we can overcome this difficulty by replacing ϵ1 with a positive parameter t and by taking ϵ2=γ(t), where γ is a function from a right neighbourhood of [math] to ]0,δ2[ such that the limits
[TABLE]
exist finite in [0,δ2[ and [0,+∞[, respectively. Then we investigate the first and second term in the asymptotic expansion of ut,γ(t) as t→0+. We observe that we have to distinguish the case when limt→0+γ(t)=0 from the case when limt→0+γ(t)>0. We shall also need the following technical lemma.
Lemma 7.3**.**
Let γ0∈]0,δ2[. Let cγ0∈R be defined by
[TABLE]
Then cγ0=0.
Proof.
By (37) we have
[TABLE]
where ρ~1,ρ~2∈C0,α(∂Ω~(ϵ2)) are defined as in (13). By Proposition 3.3, ρ~1−ρ~2 belongs to Ker(−21IΩ~(γ0)+WΩ~(γ0)∗). Then, the jump formula for vΩ~(γ0)+[ρ~1−ρ~2] in (7) implies that vΩ~(γ0)[ρ~1−ρ~2] is constant on clΩ1(0,γ0) and on clΩ2(0,γ0). It follows that cγ0=0 only if vΩ~(γ0)[ρ~1−ρ~2] equals the same constant on clΩ1(0,γ0) and on clΩ2(0,γ0). That is, if vΩ~(γ0)[ρ~1−ρ~2] is constant on the whole of clΩ~(γ0). Then we observe that by Proposition 3.3 we also have
[TABLE]
Thus ∫Ω~(γ0)ρ~1−ρ~2dσ=0 and \tilde{\rho}_{1}-\tilde{\rho}_{2}\in\bigl{(}\mathrm{Ker}(-\frac{1}{2}I_{\tilde{\Omega}(\gamma_{0})}+W^{*}_{\tilde{\Omega}(\gamma_{0})})\bigr{)}_{0}. So, by Lemma 2.2 (vii), we deduce that cγ0=0 only for ρ~1=ρ~2. However, the latter equality is in contradiction with (67). Thus cγ0=0.
∎
We now prove our main result on the asymptotic behaviour of ut,γ(t) as t→0+.
Proposition 7.4**.**
Let ΩM be an open subset of Ωo such that 0∈/clΩM. Let δM∈]0,δ1] be such that clΩM∩clΩk(ϵ1,ϵ2)=∅ for all (ϵ1,ϵ2)∈]−δM,δM[×]−δ2,δ2[ and for all k∈{1,2}. Let δM∗∈]0,δM]. Let γ be a function from ]0,δM∗[ to ]0,δ2[ such that the limits in (66)
exist finite in [0,δ2[ and [0,+∞[, respectively. Then the following statements hold:
- (i)
If γ0=0, then we have
[TABLE]
as t→0+.
2. (ii)
If γ0∈]0,δ2[, then λ0=1 and
[TABLE]
as t→0+.
Proof.
We first prove (i). If γ0=0, then we have
[TABLE]
(cf. (63)). Then the validity of (i) follows by Proposition 7.2, by the membership of tγ(t), t/logγ(t), and 1/(log(tγ(t))logγ(t)) in o(1/log(tγ(t))), and by a straightforward computation. We now pass to consider (ii). First we observe that the condition γ0∈]0,δ2[ readily implies that
λ0=1.
Then, by (63) we deduce that
[TABLE]
Thus, (61) implies that
[TABLE]
Next, by (59) we verify that
[TABLE]
for all x∈clΩM, with
dγ0≡(HΩ~(γ0)1,1−HΩ~(γ0)1,2+HΩ~(γ0)2,1−HΩ~(γ0)2,2).
By (54), (59), (61), and by equality
[TABLE]
we compute that
[TABLE]
for all x∈clΩM.
Now, the validity of (68) follows by (65), by (69)–(71), and by the asymptotic formula
[TABLE]
∎
We observe that the factor
[TABLE]
appearing in (68) vanishes when
[TABLE]
a condition which is equivalent to ∫Ω2(1,γ0)νΩ2(1,γ0)⋅∇u~dσ=0, because
[TABLE]
It also vanishes for
[TABLE]
i.e. for
[TABLE]
Condition (72) concerns u~ and thus depends on the geometry of the holes and on the boundary data f1 and f2. It is verified for example when Ω2=−Ω1 and f2(x)=f1(−x) for all x∈∂Ω2.
Instead, ρ~1 and ρ~2 depend only on the geometry of the holes (see Proposition 3.3). Accordingly, (73) is a geometric conditions on the holes. A simple arguments shows that it is verified for example when Ω2=−Ω1.
To conclude, we observe that an analog of Proposition 7.4 can also be proved for the microscopic behaviour of the solution near the boundaries of the holes. Then one can exploit such results to investigate the asymptotic behaviour of functionals of the solution. For example, one may consider the energy integral, which plays an important role in the so-called ‘topological optimization’ (cf. Novotny and J. Sokołowski [31]). The study of the energy integral also allows to investigate the capacity and then to deduce asymptotic expansions for the eigenvalues of the Dirichlet Laplacian in perforated domains (see, e.g., Courtois [10] and Abatangelo, Felli, Hillairet, and Léna [1]). The authors plan to present a detailed analysis on this subject in forthcoming papers.
Acknowledgement
M. Dalla Riva and P. Musolino acknowledge the support of ‘Progetto di Ateneo: Singular perturbation problems for differential operators – CPDA120171/12’ - University of Padova. M. Dalla Riva also acknowledges the support of HORIZON 2020 MSC EF project FAANon (grant agreement MSCA-IF-2014-EF - 654795) at the University of Aberystwyth, UK. P. Musolino also acknowledges the support of ‘INdAM GNAMPA Project 2015 - Un approccio funzionale analitico per problemi di perturbazione singolare e di omogeneizzazione’ and of an INdAM Research Fellowship. Part of the work has been carried out while P. Musolino was visiting the ‘Département de mathématiques et applications’ of the ‘École normale supérieure, Paris’. P. Musolino wishes to thank the ‘Département de mathématiques et applications’ and in particular V. Bonnaillie-Noël for the kind hospitality. P. Musolino is a Sêr CYMRU II COFUND fellow, also supported by the ‘Sêr Cymru National Research Network for Low Carbon, Energy and Environment’.
