ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS IN A PLANAR DOMAIN

. We consider a perforated domain Ω( (cid:15) ) of R 2 with a small hole of size (cid:15) and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω( (cid:15) ) as the size (cid:15) of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear (cid:15) -dependent Robin condition may degenerate into a Neumann condition for (cid:15) = 0 and the Robin datum may diverge to inﬁnity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as (cid:15) tends to 0 and to understand how the boundary condition aﬀects the behavior of the solutions when (cid:15) is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3.


Introduction
In this article we continue the analysis of [36], where we have studied the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in R n , n ≥ 3, with a nonlinear Robin boundary condition degenerating into a Neumann condition on the boundary of the small hole.
The problem considered in [36] was degenerating under three aspects: in the limit case the Robin boundary condition may degenerate into a Neumann boundary condition, the Robin datum may tend to infinity, and, finally, the size of the small hole where we consider the Robin condition tends to 0. The analysis of [36] was confined to the case of dimension n ≥ 3, since the two-dimensional case requires a different treatment.Indeed the technique of [36] is based on potential theory, and as it happens often with such method, the case of dimension n = 2 and the one of dimension n ≥ 3 need to be treated separately because of the different aspect of the fundamental solution of the Laplacian.
As already mentioned, another feature of the problem considered in the present paper and in [36] is the fact that the degenerating boundary condition is posed on the boundary of a small hole.Boundary value problems in domain with small holes have been studied by many authors.Asymptotic analysis techniques have been used for example in the works of Ammari and Kang [1], Il'in [17], Maz'ya, Movchan, and Nieves [24,25,26,27,28], Maz'ya, Nazarov, and Plamenevskij [29,30], Nieves [38], Nieves and Movchan [37], Novotny and Soko lowski [39].We also note that in Grossi and Luo [15] boundary value problems in domains with small holes have been studied with the goal of analyzing critical points of solutions.
The method of the present paper is instead, as in [36], the Functional Analytic Approach proposed by Lanza de Cristoforis in [19] for the analysis of singular perturbation problems in perforated domains.The purpose of the method is to represent the solution of a pertubed problem in terms of real analytic maps and known functions of the perturbation parameters.In particular, we observe that such method has been successfully used for example in Dalla Riva and Lanza de Cristoforis [5,6,7,8] and Lanza de Cristoforis [20], for the analysis of nonlinear boundary value problems.
In scientific and engineering practice, Robin boundary condition has an important role in many applications.Perhaps the most common use are the transport PDEs utilized in the systems such as convective-dispersive solute transport (van Genuchten and Alves [42]), heat transfer (e.g.temperature dependent boundary conditions in forming of the glass containers ass seen at [12]), and convectivediffusive mass transfer of different species.Here, the ability to define arbitrary size of the internal perturbation with Robin boundary condition is important when assessing processes at different scales -for example when analyzing sand fines migration from or into the well during oil or gas production the size of the perturbation δ (wellbore diameter) will be finite at the wellbore scale assessment but δ → 0 for field scale analysis (see e.g.[13] for various Oil and Gas applications).In [35,36] and in the present paper, we have considered a Robin problem as simplified model for the transmission problem for a composite domain with imperfect conditions along the joint boundary.Such nonlinear transmission conditions frequently appear in practical applications for various nonlinear multiphysics problems (e.g., [32,33,34]).
We begin by introducing the geometry of our problem.Therefore, we fix a regularity parameter α ∈]0, 1[ and we take two subsets, one representing the unperturbed domain Ω o and another representing the shape of the hole ω i .The sets Ω o and ω i satisfy the assumption We refer to Gilbarg and Trudinger [14] for the definition of sets and functions of the Schauder class C k,α (k ∈ N).We set If ∈]0, 0 [, then the set ω i is contained in Ω o .We think of ω i as a hole and we remove it from the unperturbed domain.Hence, we introduce the perforated domain Ω( ) by setting As the parameter tends to 0, the perforated set Ω( ) degenerates to the punctured domain Ω o \ {0}.
As we have done in [36], for each ∈]0, 0 [ we study a nonlinear boundary value problem for the Laplace operator: we consider a Neumann condition on ∂Ω o and a nonlinear Robin condition on ∂ω i .In order to define the boundary value problem in the set Ω( ), we fix two functions Next we take a family {F } ∈]0, 0 [ of functions from R to R, and two functions δ(•) and ρ(•) from ]0, 0 [ to ]0, +∞[.Now for each ∈]0, 0 [ we consider the following boundary value problem: where ν Ω o and ν ω i denote the outward unit normal to ∂Ω o and to ∂( ω i ), respectively.
As in in [36], our aim is to analyze the behavior of the solutions to problem (1.1) as → 0 and to understand how the size of the hole and the functions δ and ρ that intervene in the nonlinear Robin condition affect the asymptotic behavior of solutions to problem (1.1).We will adapt the techniques of [36] for the case of dimension n ≥ 3 to the planar perforated domain of the present paper.
The article is organized as follows.In Section 2 we analyze a toy problem in an annular domain.In Section 3 we transform problem (1.1) into an equivalent system of integral equations.In Section 4, we analyze such system and we prove our main results on the asymptotic behavior of a family of solutions and of the corresponding energy integrals.Finally, Section 5 contains some remarks on the linear case and Section 6 some conclusions.
On the other hand, in the unperturbed domain B 2 (0, 1) the Neumann problem 2) has the one-dimensional space of constant functions in B 2 (0, 1) as the space of solutions, whereas if instead we have that a = 0, then problem (2.2) does not have any solution.
As a consequence, if the compatibility condition (2.3) does not hold, the unique solution u of problem (2.1) clearly cannot converge to a solution of (2.2) as → 0 (since problem (2.2) has no solutions).Also, as we shall see, the solutions may diverge as → 0 even if a = 0, because of the terms δ( ) and ρ( ).Here we wish to investigate how the Robin condition on the region ∂ω i influences the asymptotic behavior of the solution as → 0. Now, our goal is to explicitly construct the solution u of our toy problem (2.1) and then analyze the behavior of u as → 0. We search for the solution u in the form u (x) and we need to determine the constants A and B so that the boundary conditions of problem (2.1) hold.Since ∇u (x) = A x |x| 2 , to satisfy the Neumann condition on ∂B 2 (0, 1), we must have On the other hand, to satisfy the Robin condition on ∂B 2 (0, ), we need to determine B so that i.e., Therefore, and, as a consequence, also We can rewrite (2.5) as This, for example, implies that if This means that, under suitable assumptions on the behavior of δ( ) and ρ( ) as → 0, the value of the solution u (x) at a fixed point x ∈ Ω \ {0} behaves like (a − br 0 − al 0 )/( δ( )).If instead for each positive and small enough, we take x such that |x | = , then In particular, if a − br 0 = 0 , then the value u (x ) of the solution at x is asymptotic to (a − br 0 )/( δ( )) as → 0.
We now consider the energy integral of u .A direct computation shows that We note that equation (2.5) provides a solution of the linear toy problem (2.1) also if δ( ) < 0. In case δ( ) < 0, uniqueness for the solution of problem (2.1) may fail since indeed σ = −δ( ) could be a mixed Steklov-Neumann eigenvalue of problem A detailed discussion on how to extends the results also to the case δ( ) < 0 and the analysis of the behavior of Steklov-Neumann eigenvalues may be the subject of future investigations.It is also interesting to look at a nonlinear toy problem with arbitrary functions F (•). Then repeating the same line of reasoning, the only difference appears in the equation (2.4), that will take the form If we additionally assume that the functions F : R → R are invertible then by (2.6) for each ∈]0, 1[ the constant B can be uniquely found, and the analysis can be performed in a similar way as we have previously done.However, if the functions F are not bijections, then the analysis of the existence (and possibly uniqueness) of the solution becomes more complex.For example, a solutions can be derived under specific conditions on the parameters if suitable rescaling of the functions F are locally invertible.This shows how rich the problem is even in the simple situation of a circular annular domain.On the other hand, many of the features mentioned here are preserved for the general 2D case.Below, we provide an accurate analysis of the general problem formulated above, making, where appropriate, a reference to the similar feature highlighted here for the toy problem.

Integral equation formulation of the boundary value problem
As in [35,36], we use the Functional Analytic Approach, introduced by Lanza de Cristoforis in [19], to analyze problem (1.1) when the parameter is close to 0. We refer to [9] for a detailed presentation of the method.In order to apply such approach, we need to define classical objects of potential theory.We first denote by S 2 the fundamental solution of the Laplace operator, i.e. the function from R 2 \ {0} to R defined by By means of S 2 , we construct the single layer potentials, that we use to represent the solutions of problem (1.1).So let Ω be a bounded open connected subset of R 2 of class C 1,α .We introduce the single layer potential by and the function v The normal derivative of the single layer potential on ∂Ω, instead, presents a jump.To describe such jump, we set where ν Ω denotes the outward unit normal to ∂Ω.If µ ∈ C 0,α (∂Ω), the function W * [∂Ω, µ] belongs to C 0,α (∂Ω) and we have We will use density functions with zero integral mean and thus we find it convenient to set By arguing as in [36, §3], we are ready to establish in Proposition 3.1 a correspondence between the solutions of problem (1.1) and those of a (nonlinear) system of integral equations.
to the set of those functions u ∈ C 1,α (Ω( )) which solve problem (1.1), which takes a triple (µ o , µ i , ξ) to the function is a bijection.
In the following proposition, we study the solvability of the system of integral equations (4.1)-(4.2),by applying the Implicit Function Theorem to Λ, under suitable assumptions on the partial derivative ∂ τ F l0 2π ∂Ω o g o dσ + ξ, η 0 .The symbol ∂ τ F (τ, η) denotes the partial derivative of F with respect to the first variable.
for all ∈]0, e [.Moreover, Proof.We set for all ∈]0, 2 [.Then we take U M and M as in Theorem 4.3, with Similarly, if U m and m are as in Theorem 4.4, with Therefore, we set e ≡ min{ M , m } and

Remarks on the linear case
In this section, we make further considerations on the asymptotic behavior of the solution in the linear case as the parameter tends to 0. Clearly, we can apply the results of Section 3 to the linear case.In particular, if we have In particular, All the assumptions in Sections 3 and 4 are satisfied.In particular, the solutions of the corresponding limiting system exist and are unique (see assumption (4.10)).In the linear case, equations ( 4

. 3 ) 4 .
Analytic representation formulas for the solution of the boundary value problem Under the additional assumption (3.3), we can rewrite the set of equations (3.1)-(3.2) as

Theorem 4 . 6 .
Let the assumptions of Proposition 4.1 hold.Let ũM and ũm be as in Theorem 4.3 and Theorem 4.4, respectively.Then there exist e ∈]0, 2 [ and two real analytic maps E 1 and E 2 from ] − e , e [×U to R such that Ω( )