Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 57, pp. 1–21. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2023.57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS IN A PLANAR DOMAIN PAOLO MUSOLINO, MARTIN DUTKO, GENNADY MISHURIS Abstract. We consider a perforated domain Ω(ε) of R2 with a small hole of size ε and we study the behavior of the solution of a mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal is to analyze the asymptotic behavior of the solutions to the problem as ε tends to 0 and to understand how the boundary condition affects the behavior of the solutions when ε is close to 0. The present paper extends to the planar case the results of [36] dealing with the case of dimension n ≥ 3. 1. Introduction In this article we continue the analysis of [36], where we have studied the as- ymptotic behavior of the solutions of a boundary value problem for the Laplace equation in a perforated domain in Rn, 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 ε 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. Boundary value problems with degenerating or perturbed boundary conditions have been analyzed by many authors. Here we mention, for example, Wendland, Stephan, and Hsiao [43], Kirsch [18], Costabel and Dauge [4], Ammari and Nédélec [2], Schmidt and Hiptmair [40], and [35, 36]. 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 2020 Mathematics Subject Classification. 35J25, 31B10, 35B25, 35C20, 47H30. Key words and phrases. Singularly perturbed boundary value problem; Laplace equation; nonlinear Robin condition; perforated planar domain; integral equation. ©2023. This work is licensed under a CC BY 4.0 license. Submitted February 15, 2023. Published September 11, 2023. 1 2 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 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 impor- tant 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 convective- diffusive 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 mi- gration 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 ap- pear 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 un- perturbed domain Ωo and another representing the shape of the hole ωi. The sets Ωo and ωi satisfy the assumption ωi and Ωo are bounded open connected subsets of R2 of class C1,α such that 0 ∈ Ωo ∩ ωi and that R2 \ ωi and R2 \Ωo are connected. We refer to Gilbarg and Trudinger [14] for the definition of sets and functions of the Schauder class Ck,α (k ∈ N). We set ε0 ≡ sup{θ ∈ ]0,+∞[ : εωi ⊆ Ωo, ∀ε ∈ ]−θ, θ[} . If ε ∈]0, ε0[, then the set εωi is contained in Ωo. We think of εωi as a hole and we remove it from the unperturbed domain. Hence, we introduce the perforated domain Ω(ε) by setting Ω(ε) ≡ Ωo \ εωi ∀ε ∈ ]0, ε0[ . As the parameter ε tends to 0, the perforated set Ω(ε) degenerates to the punctured domain Ωo \ {0}. EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 3 As we have done in [36], for each ε ∈]0, ε0[ we study a nonlinear boundary value problem for the Laplace operator: we consider a Neumann condition on ∂Ωo and a nonlinear Robin condition on ε∂ωi. In order to define the boundary value problem in the set Ω(ε), we fix two functions go ∈ C0,α(∂Ωo) , gi ∈ C0,α(∂ωi) . Next we take a family {Fε}ε∈]0,ε0[ of functions from R to R, and two functions δ(·) and ρ(·) from ]0, ε0[ to ]0,+∞[. Now for each ε ∈]0, ε0[ we consider the following boundary value problem: ∆u(x) = 0 ∀x ∈ Ω(ε) , ∂ ∂νΩo u(x) = go(x) ∀x ∈ ∂Ωo , ∂ ∂νεωi u(x) = δ(ε)Fε(u(x)) + gi(x/ε) ρ(ε) ∀x ∈ ε∂ωi , (1.1) where νΩo and νεωi denote the outward unit normal to ∂Ωo and to ∂(εωi), respec- tively. As in in [36], our aim is to analyze the behavior of the solutions to problem (1.1) as ε→ 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. 2. A toy problem As we have done in [35, 36], we consider problem (1.1) in the annular domain Ω(ε) ≡ B2(0, 1) \ B2(0, ε) , where, for r > 0, the symbol B2(0, r) denotes the open ball in R2 of center 0 and radius r. In other words, we take Ωo ≡ B2(0, 1) and ωi ≡ B2(0, 1). We set ε0 = 1, Fε(τ) = τ for all τ ∈ R and for all ε ∈]0, ε0[, go = a, and gi = b, where a, b ∈ R. In addition, we take two functions δ, ρ : ]0, 1[7→]0,+∞[ and for each ε ∈]0, 1[ we consider the problem ∆u(x) = 0 ∀x ∈ B2(0, 1) \ B2(0, ε) , ∂ ∂νB2(0,1) u(x) = a ∀x ∈ ∂B2(0, 1) , ∂ ∂νB2(0,ε) u(x) = δ(ε)u(x) + b ρ(ε) ∀x ∈ ∂B2(0, ε) . (2.1) It is well known that for each ε ∈]0, 1[ problem (2.1) has a unique solution in C1,α(Ω(ε)). We denote such a solution by uε. 4 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 On the other hand, in the unperturbed domain B2(0, 1) the Neumann problem ∆u(x) = 0 ∀x ∈ B2(0, 1) , ∂ ∂νB2(0,1) u(x) = a ∀x ∈ ∂B2(0, 1) (2.2) is subject to compatibility conditions on the Neumann datum on ∂B2(0, 1). In particular, in this specific case of constant Neumann datum, problem (2.2) has a solution if and only if a = 0 . (2.3) For a = 0, the Neumann problem (2.2) has the one-dimensional space of constant functions in B2(0, 1) as the space of solutions, whereas if instead we have that a 6= 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 uε of problem (2.1) clearly cannot converge to a solution of (2.2) as ε→ 0 (since problem (2.2) has no solutions). Also, as we shall see, the solutions may diverge as ε→ 0 even if a = 0, because of the terms δ(ε) and ρ(ε). Here we wish to investigate how the Robin condition on the region ε∂ωi influences the asymptotic behavior of the solution as ε→ 0. Now, our goal is to explicitly construct the solution uε of our toy problem (2.1) and then analyze the behavior of uε as ε→ 0. We search for the solution uε in the form uε(x) ≡ Aε log |x|+Bε ∀x ∈ Ω(ε) , and we need to determine the constants Aε and Bε so that the boundary conditions of problem (2.1) hold. Since ∇uε(x) = Aε x |x|2 , to satisfy the Neumann condition on ∂B2(0, 1), we must have Aε = a . On the other hand, to satisfy the Robin condition on ∂B2(0, ε), we need to determine Bε so that x |x| · a x |x|2 = δ(ε)(a log |x|+Bε) + b ρ(ε) ∀x ∈ ∂B2(0, ε) , (2.4) i.e., a ε = δ(ε)(a log ε+Bε) + b ρ(ε) ∀x ∈ ∂B2(0, ε) . Therefore, Bε = 1 δ(ε) (a ε − b ρ(ε) ) − a log ε , and, as a consequence, also uε(x) ≡ a log |x|+ 1 δ(ε) (a ε − b ρ(ε) ) − a log ε ∀x ∈ Ω(ε) . (2.5) We can rewrite (2.5) as uε(x) ≡ a log |x|+ 1 εδ(ε) ( a− b ε ρ(ε) − aεδ(ε) log ε ) ∀x ∈ Ω(ε) . EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 5 This, for example, implies that if l0 ≡ lim ε→0 εδ(ε) log ε ∈ R , r0 ≡ lim ε→0 ε ρ(ε) ∈ R , and a− br0 − al0 6= 0 , then the value of the solution uε(x) is asymptotic to (a − br0 − al0)/(εδ(ε)) as ε tends to 0 for all fixed x ∈ Ω \ {0}. This means that, under suitable assumptions on the behavior of δ(ε) and ρ(ε) as ε → 0, the value of the solution uε(x) at a fixed point x ∈ Ω \ {0} behaves like (a− br0 − al0)/(εδ(ε)). If instead for each ε positive and small enough, we take x̃ε such that |x̃ε| = ε, then uε(x̃ε) = a log ε+ 1 εδ(ε) ( a− b ε ρ(ε) − aεδ(ε) log ε ) = 1 εδ(ε) ( aεδ(ε) log ε+ a− b ε ρ(ε) − aεδ(ε) log ε ) = 1 εδ(ε) ( a− b ε ρ(ε) ) . In particular, if a− br0 6= 0 , then the value uε(x̃ε) of the solution at x̃ε is asymptotic to (a − br0)/(εδ(ε)) as ε→ 0. We now consider the energy integral of uε. A direct computation shows that∫ Ω(ε) |∇uε(x)|2 dx = ∫ Ω(ε) |∇ ( a log |x| ) |2 dx = ∫ Ω(ε) a2 1 |x|2 dx = a22π ∫ 1 ε 1 r dr = a22π ( − log ε ) . We note that equation (2.5) provides a solution of the linear toy problem (2.1) also if δ(ε) < 0. In case δ(ε) < 0, uniqueness for the solution of problem (2.1) may fail since indeed σ = −δ(ε) could be a mixed Steklov-Neumann eigenvalue of problem ∆u = 0 in Ω(ε) , ∂ ∂νΩ(ε) u = 0 on ∂B2(0, 1) , ∂ ∂νΩ(ε) u = σu on ∂B2(0, ε) . A detailed discussion on how to extends the results also to the case δ(ε) < 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 Fε(·). Then repeating the same line of reasoning, the only difference appears in the 6 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 equation (2.4), that will take the form x |x| · a x |x|2 = δ(ε)Fε(a log |x|+Bε) + b ρ(ε) ∀x ∈ ∂B2(0, ε) . (2.6) If we additionally assume that the functions Fε : R→ R are invertible then by (2.6) for each ε ∈]0, 1[ the constant Bε can be uniquely found, Bε = F−1 ε (aρ(ε)− εb ερ(ε)δ(ε) ) − a log ε , and the analysis can be performed in a similar way as we have previously done. However, if the functions Fε 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 Fε 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. 3. 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 ε 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 S2 the fundamental solution of the Laplace operator, i.e. the function from R2 \ {0} to R defined by S2(x) ≡ 1 2π log |x| ∀x ∈ R2 \ {0} . By means of S2, 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 R2 of class C1,α. We introduce the single layer potential by v[∂Ω, µ](x) ≡ ∫ ∂Ω S2(x− y)µ(y) dσy ∀x ∈ R2 , for all µ ∈ C0(∂Ω). If µ ∈ C0(∂Ω), then v[∂Ω, µ] is continuous in R2. Moreover, if µ ∈ C0,α(∂Ω), then the function v+[∂Ω, µ] ≡ v[∂Ω, µ]|Ω belongs to C1,α(Ω), and the function v−[∂Ω, µ] ≡ v[∂Ω, µ]|R2\Ω belongs to C1,α loc (R2 \ Ω). The normal derivative of the single layer potential on ∂Ω, instead, presents a jump. To describe such jump, we set W ∗[∂Ω, µ](x) ≡ ∫ ∂Ω νΩ(x) · ∇S2(x− y)µ(y) dσy ∀x ∈ ∂Ω , where νΩ denotes the outward unit normal to ∂Ω. If µ ∈ C0,α(∂Ω), the function W ∗[∂Ω, µ] belongs to C0,α(∂Ω) and we have ∂ ∂νΩ v±[∂Ω, µ] = ∓1 2 µ+W ∗[∂Ω, µ] on ∂Ω . EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 7 We will use density functions with zero integral mean and thus we find it conve- nient to set C0,α(∂Ωo)0 ≡ { f ∈ C0,α(∂Ωo) : ∫ ∂Ωo f dσ = 0 } . By arguing as in [36, §3], we are ready to establish in Proposition 3.1 a corre- spondence between the solutions of problem (1.1) and those of a (nonlinear) system of integral equations. Proposition 3.1. Let ε ∈]0, ε0[. Then the map from the set of triples (µo, µi, ξ) ∈ C0,α(∂Ωo)0 × C0,α(∂ωi)× R such that − 1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + ∫ ∂ωi νΩo(x) · ∇S2(x− εs)µi(s) dσs = go(x) ∀x ∈ ∂Ωo , (3.1) 1 2 µi(t) + ε ∫ ∂Ωo νωi(t) · ∇S2(εt− y)µo(y) dσy + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = εδ(ε)Fε (∫ ∂Ωo S2(εt− y)µo(y) dσy + ∫ ∂ωi S2(t− s)µi(s) dσs + log ε 2π ∫ ∂ωi µi dσ + ξ δ(ε)ε ) + gi(t) ε ρ(ε) ∀t ∈ ∂ωi , (3.2) to the set of those functions u ∈ C1,α(Ω(ε)) which solve problem (1.1), which takes a triple (µo, µi, ξ) to the function∫ ∂Ωo S2(x− y)µo(y) dσy + ∫ ∂ωi S2(x− εs)µi(s) dσs + ξ δ(ε)ε ∀x ∈ Ω(ε) is a bijection. By Proposition 3.1 we can study the behavior of the solutions of boundary value problem (1.1) by analyzing those of the system of integral equations (3.1)-(3.2) as ε → 0. As we have done in [36], we make some structural assumptions on the nonlinearity and we assume that there exist ε1 ∈]0, ε0[, m ∈ N, a real analytic function F̃ from Rm+1 to R, and a function η(·) from ]0, ε1[ to Rm such that η0 ≡ lim ε→0 η(ε) ∈ Rm and that εδ(ε)Fε ( 1 εδ(ε) τ ) = F̃ (τ, η(ε)) for all (τ, ε) ∈ R×]0, ε1[. (3.3) 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 − 1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + ∫ ∂ωi νΩo(x) · ∇S2(x− εs)µi(s) dσs = go(x) ∀x ∈ ∂Ωo , (4.1) 8 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 1 2 µi(t) + ε ∫ ∂Ωo νωi(t) · ∇S2(εt− y)µo(y) dσy + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = F̃ ( εδ(ε) ∫ ∂Ωo S2(εt− y)µo(y) dσy + εδ(ε) ∫ ∂ωi S2(t− s)µi(s) dσs + εδ(ε) log ε 2π ∫ ∂ωi µi dσ + ξ, η(ε) ) + gi(t) ε ρ(ε) ∀t ∈ ∂ωi , (4.2) for all ε ∈]0, ε1[. In order to pass to the limit as ε → 0 in equations (4.1)-(4.2), we need to know the asymptotic behavior for ε close to 0 of the quantities εδ(ε), εδ(ε) log ε, and ε ρ(ε) which appear in (4.2). As a consequence, we now assume that l0 ≡ lim ε→0 εδ(ε) log ε ∈ R , r0 ≡ lim ε→0 ε ρ(ε) ∈ R . (4.3) Condition (4.3) implies also limε→0 εδ(ε) = 0. In (4.1)-(4.2), we replace the quantities εδ(ε) , εδ(ε) log ε , η(ε) , ε ρ(ε) , by the auxiliary variables γ1, γ2, γ3, and γ4,respectively, and we introduce the operator Λ ≡ (Λo,Λi) from ] − ε1, ε1[×Rm+3 × C0,α(∂Ωo)0 × C0,α(∂ωi) × R to C0,α(∂Ωo)× C0,α(∂ωi) by setting Λo[ε, γ1, γ2, γ3, γ4, µ o, µi, ξ](x) ≡ −1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + ∫ ∂ωi νΩo(x) · ∇S2(x− εs)µi(s) dσs − go(x) ∀x ∈ ∂Ωo , (4.4) Λi[ε, γ1, γ2, γ3, γ4, µ o, µi, ξ](t) ≡ 1 2 µi(t) + ε ∫ ∂Ωo νωi(t) · ∇S2(εt− y)µo(y) dσy + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs − F̃ ( γ1 ∫ ∂Ωo S2(εt− y)µo(y) dσy + γ1 ∫ ∂ωi S2(t− s)µi(s) dσs + γ2 2π ∫ ∂ωi µi dσ + ξ, γ3 ) − gi(t)γ4 ∀t ∈ ∂ωi , (4.5) for all (ε, γ1, γ2, γ3, γ4, µ o, µi, ξ) ∈] − ε1, ε1[×Rm+3 × C0,α(∂Ωo)0 × C0,α(∂ωi) × R. By definitions (4.4)-(4.5), for ε ∈]0, ε1[ the system of equations Λo[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) , µo, µi, ξ](x) = 0 ∀x ∈ ∂Ωo , (4.6) Λi[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) , µo, µi, ξ](t) = 0 ∀t ∈ ∂ωi (4.7) EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 9 is equivalent to the system of integral equations (4.1)-(4.2). Letting ε → 0 in (4.6)-(4.7), we obtain the equations − 1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + νΩo(x) · ∇S2(x) ∫ ∂ωi µi(s) dσs = go(x) ∀x ∈ ∂Ωo , (4.8) 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = F̃ ( l0 2π ∫ ∂ωi µi dσ + ξ, η0 ) + gi(t)r0 ∀t ∈ ∂ωi . (4.9) For ε ∈]0, ε1[ small enough, we would like to prove the existence of solutions (µo, µi, ξ) to (4.6)-(4.7) around a solution of the limiting system (4.8)-(4.9). There- fore, we further assume that System (4.8)-(4.9) in the unknown (µo, µi, ξ) admits a solution (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0 × C0,α(∂ωi)× R. (4.10) We now note that if (µ̃o, µ̃i, ξ̃) is a solution of the system (4.8)-(4.9), by integrating (4.8) on ∂Ωo and by the equalities∫ ∂Ωo ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µ̃o(y) dσy dσx = 1 2 ∫ ∂Ωo µ̃o(y) dσy (cf. [9, Lemma 6.11]) and ∫ ∂Ωo νΩo(x) · ∇S2(x) dσx = 1 (cf. [9, Corollary 4.6]), we must have∫ ∂ωi µ̃i(s) dσs = ∫ ∂Ωo go(x) dσx . This implies that the triple (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0 × C0,α(∂ωi)× R is a solution of system (4.11)–(4.13) below − 1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy = go(x)− νΩo(x) · ∇S2(x) ∫ ∂Ωo go(y) dσy ∀x ∈ ∂Ωo , (4.11) 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ, η0 ) + gi(t)r0 ∀t ∈ ∂ωi , (4.12) ∫ ∂ωi µi(s) dσs = ∫ ∂Ωo go(x) dσx . (4.13) By [9, Theorem 6.25], we deduce that there exists a unique solution µ̃o in C0,α(∂Ωo)0 of (4.11). In other words, if there exists a solution (µ̃o, µ̃i, ξ̃) of the system (4.8)-(4.9), then µ̃o is determined as the unique solution in C0,α(∂Ωo)0 of (4.11). 10 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 To have a pair (µ̃i, ξ̃) in C0,α(∂ωi)× R solving (4.12)-(4.13), we observe that if there exists ξ̃ ∈ R such that∫ ∂Ωo go(x) dσx = |∂ωi|1F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) + ∫ ∂ωi gi(t) dσtr0 (4.14) then [9, Corollary 6.15] implies the existence of a unique solution µ̃i in C0,α(∂ωi) of 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) + gi(t)r0 ∀t ∈ ∂ωi , and such solution satisfies also∫ ∂ωi µ̃i(s) dσs = ∫ ∂Ωo go(x) dσx . In other words this means that if there exists ξ̃ ∈ R such that equation (4.14) holds, then there exists a unique pair (µ̃o, µ̃i) in C0,α(∂Ωo)0 × C0,α(∂ωi) such that the triple (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0 × C0,α(∂ωi)× R solves system (4.8)-(4.9). Furthermore, we note that equality (4.14) can be rewritten as F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) = 1 |∂ωi|1 ( r0 ∫ ∂ωi gi dσ − ∫ ∂Ωo go dσ ) . (4.15) Thus, in particular, if F̃ (·, η0) is not globally invertible, there can be multiple ξ̃ ∈ R such that (4.15) holds. 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 suit- able assumptions on the partial derivative ∂τ F̃ ( l0 2π ∫ ∂Ωo g o dσ + ξ̃, η0 ) . The symbol ∂τ F̃ (τ, η) denotes the partial derivative of F̃ with respect to the first variable. Proposition 4.1. Let assumptions (3.3) and (4.3) hold. Let (µ̃o, µ̃i, ξ̃) be as in assumption (4.10). Assume that ∂τ F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) 6= 0 . Then there exist ε2 ∈]0, ε1[, an open neighborhood U of (0, l0, η0, r0) in Rm+3, an open neighborhood V of (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0×C0,α(∂ωi)×R, and a real analytic map (Mo,M i,Ξ) from ]− ε2, ε2[×U to V such that( εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ) ∈ U ∀ε ∈]0, ε2[ , and such that the set of zeros of Λ in ]− ε2, ε2[×U × V coincides with the graph of (Mo,M i,Ξ). In particular,( Mo[0, 0, l0, η0, r0],M i[0, 0, l0, η0, r0],Ξ[0, 0, l0, η0, r0] ) = (µ̃o, µ̃i, ξ̃) . Proof. Standard results of classical potential theory (see, e.g., [9], Miranda [31], Lanza de Cristoforis and Rossi [22]), real analyticity results for integral operators with real analytic kernel [21], assumption (3.3) and real analyticity results for the composition operator ([3, p. 10], [16], and Valent [41, Thm. 5.2]) imply that Λ is a real analytic operator from ] − ε1, ε1[×Rm+3 × C0,α(∂Ωo)0 × C0,α(∂ωi) × R to EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 11 C0,α(∂Ωo) × C0,α(∂ωi). We verify that by standard calculus in Banach space the partial differential ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃] of Λ at (0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃) with respect to the variable (µo, µi, ξ) is delivered by ∂(µo,µi,ξ)Λ o[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃](µo, µi, ξ)(x) ≡ −1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + νΩo(x) · ∇S2(x) ∫ ∂ωi µi(s) dσs ∀x ∈ ∂Ωo , ∂(µo,µi,ξ)Λ i[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃](µo, µi, ξ)(t) ≡ 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs − ∂τ F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 )( l0 2π ∫ ∂ωi µi dσ + ξ ) ∀t ∈ ∂ωi , for all (µo, µi, ξ) ∈ C0,α(∂Ωo)0 × C0,α(∂ωi) × R. The next step is to prove that the partial differential ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃] is a homeomorphism from C0,α(∂Ωo)0 × C0,α(∂ωi) × R onto C0,α(∂Ωo) × C0,α(∂ωi). We observe that the partial differential ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃] is a Fredholm operator of index 0: indeed it is the sum of an invertible operator and a compact operator. As a consequence, to prove that the operator ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃] is a homeomorphism, it is enough to show that it is injective. Therefore, let us assume that ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃](µo, µi, ξ) = 0 . We integrate the equality ∂(µo,µi,ξ)Λ o[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃](µo, µi, ξ)(x) = 0 ∀x ∈ ∂Ωo , that together with the equalities∫ ∂Ωo ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy dσx = 1 2 ∫ ∂Ωo µo(y) dσy (cf. [9, Lemma 6.11]) and ∫ ∂Ωo νΩo(x) · ∇S2(x) dσx = 1 (cf. [9, Corollary 4.6]), implies ∫ ∂ωi µi(s) dσs = 0 . (4.16) Accordingly, −1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy = 0 ∀x ∈ ∂Ωo . Since ∫ ∂Ωo µ o dσ = 0, by [9, Theorem 6.25] we have µo = 0. Then we note that by (4.16) equality ∂(µo,µi,ξ)Λ i[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃](µo, µi, ξ)(t) = 0 ∀t ∈ ∂ωi reads as 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs − ∂τ F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) (4.17) 12 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 for all t ∈ ∂ωi. By equality (4.16), by∫ ∂ωi ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs dσt = 1 2 ∫ ∂ωi µi(s) dσs (cf. [9, Lemma 6.11]), and by integrating (4.17) on ∂ωi, we deduce that ξ = 0. Then [9, Corollary 6.15] implies that µi = 0. Hence, we have shown that the operator ∂(µo,µi,ξ)Λ[0, 0, l0, η0, r0, µ̃ o, µ̃i, ξ̃] is injective, and as a consequence, being a Fredholm operator of index 0, also a homeomorphism. Therefore, we can apply the Implicit Function Theorem for real analytic maps in Banach spaces (cf. Deimling [11, Thm. 15.3]) and deduce that there exist ε2 ∈]0, ε1[, an open neighborhood U of (0, l0, η0, r0) in Rm+3, an open neighborhood V of (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0 × C0,α(∂ωi) × R, and a real analytic map (Mo,M i,Ξ) from ] − ε2, ε2[×U to V such that ( εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ) ∈ U ∀ε ∈]0, ε2[ , and such that the set of zeros of Λ in ]− ε2, ε2[×U × V coincides with the graph of (Mo,M i,Ξ). In particular,( Mo[0, 0, l0, η0, r0],M i[0, 0, l0, η0, r0],Ξ[0, 0, l0, η0, r0] ) = (µ̃o, µ̃i, ξ̃) , and thus the proof is complete. � By Proposition 4.1 we know that there exists a family of solutions to the system of integral equations (4.1)-(4.2). Then we can exploit the representation formula of Proposition 3.1 and introduce a family of solutions to (1.1). We do so in the following Definition 4.2. Definition 4.2. Let the assumptions of Proposition 4.1 hold. Then we set u(ε, x) = ∫ ∂Ωo S2(x− y)Mo[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](y) dσy + ∫ ∂ωi S2(x− εs)M i[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](s) dσs + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε for all x ∈ Ω(ε) and all ε ∈]0, ε2[. We are ready to exploit the representation formula of Proposition 3.1 and the analyticity result of Proposition 4.1 concerning the solutions of the system of inte- gral equations (4.1)-(4.2) in order to prove formulas for suitable restrictions of the solutions u(ε, ·) and for the corresponding energy integral in terms of real analytic maps. We start by considering the restriction of the solution u(ε, ·) to a set which is “far” from the point where the hole degenerates. Theorem 4.3. Let the assumptions of Proposition 4.1 hold. Let ΩM be a bounded open subset of Ωo such that 0 6∈ ΩM . Then there exist εM ∈]0, ε2[ and a real analytic map UM from ]− εM , εM [×U to C1,α(ΩM ) such that ΩM ⊆ Ω(ε) ∀ε ∈]0, εM [ , EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 13 and that u(ε, x) = UM [ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](x) + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε (4.18) for all x ∈ ΩM and al ε ∈]0, εM [. Moreover, if we set ũM (x) ≡ ∫ ∂Ωo S2(x− y)µ̃o(y) dσy ∀x ∈ Ωo , we have that UM [0, 0, l0, η0, r0] = ũM |ΩM + S2|ΩM ∫ ∂Ωo g o dσ, and ũM is a solution of the Neumann problem ∆u(x) = 0 ∀x ∈ Ωo , ∂ ∂νΩo u(x) = go(x)− ∂ ∂νΩo S2(x) ∫ ∂Ωo go dσ ∀x ∈ ∂Ωo . (4.19) Proof. We can take εM ∈]0, ε2[ small enough so that ΩM ∩ εωi = ∅ ∀ε ∈]− εM , εM [ . Recalling Definition 4.2, we set UM [ε, γ1, γ2, γ3, γ4](x) ≡ ∫ ∂Ωo S2(x− y)Mo[ε, γ1, γ2, γ3, γ4](y) dσy + ∫ ∂ωi S2(x− εs)M i[ε, γ1, γ2, γ3, γ4](s) dσs ∀x ∈ ΩM , for all (ε, γ1, γ2, γ3, γ4) ∈]− εM , εM [×U . Then Proposition 4.1 and real analyticity results for integral operators with real analytic kernel (cf. [21]) imply that UM is a real analytic map from ]− εM , εM [×U to C1,α(ΩM ) and that equality (4.18) holds. By Proposition 4.1, we also have UM [0, 0, l0, η0, r0] = ũM |ΩM + S2|ΩM ∫ ∂Ωo g o dσ, Moreover, standard properties of the single layer potential (cf. [9, §4.4]) imply that ũM is a solution of the Neumann problem (4.19). � Then we consider the behavior of the rescaled solution u(ε, εt). Theorem 4.4. Let the assumptions of Proposition 4.1 hold. Let Zm be the real analytic map from ]− ε2, ε2[×U to R defined by Zm[ε, γ1, γ2, γ3, γ4] ≡ ∫ ∂ωi M i[ε, γ1, γ2, γ3, γ4](s) dσs , for all (ε, γ1, γ2, γ3, γ4) ∈]− ε2, ε2[×U . Let Ωm be a bounded open subset of R2 \ωi. Then there exist εm ∈]0, ε2[ and a real analytic map Um from ] − εm, εm[×U to C1,α(Ωm) such that εΩm ⊆ Ω(ε) ∀ε ∈]0, εm[ , and that u(ε, εt) = Um[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](t) + log ε 2π Zm[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε ∀t ∈ Ωm , 14 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 for all ε ∈]0, εm[. Moreover, if we set ũm(t) ≡ ∫ ∂ωi S2(t− s)µ̃i(s) dσs + ∫ ∂Ωo S2(y)µ̃o(y) dσy ∀t ∈ R2 \ ωi , we have that Um[0, 0, l0, η0, r0] = ũm|Ωm , and ũm is a solution of the Neumann problem ∆u(t) = 0 ∀t ∈ R2 \ ωi , ∂ ∂νωi u(t) = F̃ ( l0 2π ∫ ∂Ωo go dσ + ξ̃, η0 ) + gi(t)r0 ∀t ∈ ∂ωi , (4.20) and Zm[0, 0, l0, η0, r0] = ∫ ∂Ωo go dσ . Proof. We take εm ∈]0, ε2[ small enough and we can assume that εΩm ⊆ Ωo ∀ε ∈]− εm, εm[ . If ε ∈]0, εm[ then u(ε, εt) = ∫ ∂Ωo S2(εt− y)Mo[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](y) dσy + ∫ ∂ωi S2(εt− εs)M i[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](s) dσs + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε = ∫ ∂Ωo S2(εt− y)Mo[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](y) dσy + ∫ ∂ωi S2(t− s)M i[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](s) dσs + log ε 2π ∫ ∂ωi M i[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](s) dσs + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε ∀t ∈ Ωm (cf. Definition 4.2). Hence, we set Um[ε, γ1, γ2, γ3, γ4](t) ≡ ∫ ∂Ωo S2(εt− y)Mo[ε, γ1, γ2, γ3, γ4](y) dσy + ∫ ∂ωi S2(t− s)M i[ε, γ1, γ2, γ3, γ4](s) dσs ∀t ∈ Ωm , for all (ε, γ1, γ2, γ3, γ4) ∈] − εm, εm[×U . By arguing as in the proof of Theo- rem 4.3, we verify that Um and the map Zm of the statement are real analytic from ] − εm, εm[×U to C1,α(Ωm) and to R, respectively, and that equality (4.18) holds. By Proposition 4.1, we also deduce that Zm[0, 0, l0, η0, r0] = ∫ ∂Ωo g o dσ, that Um[0, 0, l0, η0, r0] = ũm|Ωm . Also by standard properties of the single layer potential (cf. [9, §4.4]), we deduce that ũm is a solution of the Neumann problem (4.20). � Remark 4.5. We note that if ∫ ∂ωi µ̃ i dσ 6= 0 (i.e., if ∫ ∂Ωo g o dσ 6= 0), then the function ũm of Theorem 4.4 is not harmonic at infinity (cf. [9, Definition 3.21 and Theorem 4.23]). EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 15 Finally, we study the behavior of the energy integral ∫ Ω(ε) |∇u(ε, x)|2 dx as the parameter ε approaches 0. 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 εe ∈]0, ε2[ and two real analytic maps E1 and E2 from ]− εe, εe[×U to R such that∫ Ω(ε) |∇u(ε, x)|2 dx = E1[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] + (log ε)E2[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] , (4.21) for all ε ∈]0, εe[. Moreover, E1[0, 0, l0, η0, r0] = ∫ ∂Ωo ( ũM (x) + S2(x) ∫ ∂Ωo go dσ ) νΩo(x) · ∇ ( ũM (x) + S2(x) ∫ ∂Ωo go dσ ) dσx − ∫ ∂ωi ũm(t)νωi(t) · ∇ũm(t) dσt (4.22) and E2[0, 0, l0, η0, r0] = − 1 2π (∫ ∂Ωo go dσ )2 . (4.23) Proof. We set cε ≡ log ε 2π Zm [ ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] + Ξ[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] δ(ε)ε ∀ε ∈]0, ε2[ . The Divergence Theorem implies that∫ Ω(ε) |∇u(ε, x)|2 dx = ∫ Ω(ε) |∇ ( u(ε, x)− cε ) |2 dx = ∫ ∂Ωo ( u(ε, x)− cε ) ∂ ∂νΩo ( u(ε, x)− cε ) dσx − ∫ ∂εωi ( u(ε, x)− cε ) ∂ ∂νεωi ( u(ε, x)− cε ) dσx = ∫ ∂Ωo ( u(ε, x)− cε ) ∂ ∂νΩo ( u(ε, x)− cε ) dσx − ∫ ∂ωi ( u(ε, εt)− cε ) νωi(t) · ∇t ( u(ε, εt)− cε ) dσt , for all ε ∈]0, ε2[. Then we take UM and εM as in Theorem 4.3, with ΩM ≡ Ωo \ B2(0, rM ) , 16 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 for some rM > 0 such that B2(0, rM ) ⊆ Ωo. We verify that if ε ∈]0, εM [, then∫ ∂Ωo ( u(ε, x)− cε ) ∂ ∂νΩo ( u(ε, x)− cε ) dσx = ∫ ∂Ωo UM [ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](x) × νΩo(x) · ∇UM [ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](x) dσx − log ε 2π Zm[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ] × ∫ ∂Ωo νΩo(x) · ∇UM [ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](x) dσx . Similarly, if Um and εm are as in Theorem 4.4, with Ωm ≡ B2(0, rm) \ ωi , for some rm > 0 such that B2(0, rm) ⊇ ωi, then if ε ∈]0, εm[,∫ ∂ωi ( u(ε, εt)− cε ) νωi(t) · ∇t ( u(ε, εt)− cε ) dσt = ∫ ∂ωi Um[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](t) × νωi(t) · ∇Um[ε, εδ(ε), εδ(ε) log ε, η(ε), ε ρ(ε) ](t) dσt . Therefore, we set εe ≡ min{εM , εm} and E1[ε, γ1, γ2, γ3, γ4] ≡ ∫ ∂Ωo UM [ε, γ1, γ2, γ3, γ4](x)νΩo(x) · ∇UM [ε, γ1, γ2, γ3, γ4](x) dσx − ∫ ∂ωi Um[ε, γ1, γ2, γ3, γ4](t)νωi(t) · ∇Um[ε, γ1, γ2, γ3, γ4](t) dσt and E2[ε, γ1, γ2, γ3, γ4] ≡ − 1 2π Zm[ε, γ1, γ2, γ3, γ4] ∫ ∂Ωo νΩo(x) · ∇UM [ε, γ1, γ2, γ3, γ4](x) dσx for all (ε, γ1, γ2, γ3, γ4) ∈]− εe, εe[×U . We verify that the maps E1 and E2 are real analytic from ]− εe, εe[×U to R and that equality (4.21) holds. Moreover, we also have E1[0, 0, l0, η0, r0] = ∫ ∂Ωo ( ũM (x) + S2(x) ∫ ∂Ωo go dσ ) νΩo(x) · ∇ ( ũM (x) + S2(x) ∫ ∂Ωo go dσ ) dσx − ∫ ∂ωi ũm(t)νωi(t) · ∇ũm(t) dσt and E2[0, 0, l0, η0, r0] EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 17 = − 1 2π ∫ ∂Ωo go dσ ∫ ∂Ωo νΩo(x) · ∇ ( ũM (x) + S2(x) ∫ ∂Ωo go dσ ) dσx = − 1 2π (∫ ∂Ωo go dσ )2 , and accordingly equalities (4.22) and (4.23) hold. � 5. 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 ε tends to 0. Clearly, we can apply the results of Section 3 to the linear case. In particular, if we have Fε(τ) = τ ∀(τ, ε) ∈ R×]0, ε0[ , problem (1.1) reduces to the linear problem ∆u(x) = 0 ∀x ∈ Ω(ε) , ∂ ∂νΩo u(x) = go(x) ∀x ∈ ∂Ωo , ∂ ∂νεωi u(x) = δ(ε)u(x) + gi(x/ε) ρ(ε) ∀x ∈ ε∂ωi . (5.1) We also know that for each ε ∈]0, ε0[, problem (5.1) has a unique solution in C1,α(Ω(ε)), which we denote by u[ε]. Clearly, εδ(ε)Fε ( 1 εδ(ε) τ ) = τ ∀(τ, ε) ∈ R×]0, ε0[ , and thus we can take for example η(ε) = 0 ∀ε ∈]0, ε0[ , F̃ (τ, η) = τ ∀(τ, η) ∈ R2 . In particular, η0 = 0 , ∂τ F̃ (τ, η) = 1 ∀(τ, η) ∈ R2 . 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 (4.8)-(4.9) become − 1 2 µo(x) + ∫ ∂Ωo νΩo(x) · ∇S2(x− y)µo(y) dσy + νΩo(x) · ∇S2(x) ∫ ∂ωi µi(s) dσs = go(x) ∀x ∈ ∂Ωo , (5.2) 1 2 µi(t) + ∫ ∂ωi νωi(t) · ∇S2(t− s)µi(s) dσs = l0 2π ∫ ∂ωi µi dσ + ξ + gi(t)r0 ∀t ∈ ∂ωi . (5.3) 18 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 By arguing as in the proof of Proposition 4.1, one can prove that the system (5.2)- (5.3) in the unknown (µo, µi, ξ) admits a unique solution (µ̃o, µ̃i, ξ̃) in C0,α(∂Ωo)0× C0,α(∂ωi)× R. In particular, by integrating (5.2), we recall that we obtain∫ ∂ωi µ̃i(s) dσs = ∫ ∂Ωo go(x) dσx . By integrating (5.3), we deduce that∫ ∂Ωo go(x) dσx = |∂ωi|1 l0 2π ∫ ∂Ωo go(x) dσx + |∂ωi|1ξ̃ + r0 ∫ ∂ωi gi(t) dσt , which implies ξ̃ = 1 |∂ωi|1 (( 1− |∂ωi|1 l0 2π )∫ ∂Ωo go(x) dσx − r0 ∫ ∂ωi gi(t) dσt ) . We note that in case Ωo = ωi = B2(0, 1) and go(x) = a ∀x ∈ ∂B2(0, 1) , gi(t) = b ∀t ∈ ∂B2(0, 1) , we obtain ξ̃ = (( 1− 2π l0 2π ) a− br0 ) = ( a− al0 − br0 ) . Therefore, we recover also the results of Section 2. 6. Conclusions In this article, we have considered a perforated domain Ω(ε) of R2 with a small hole of size ε and we have studied the behavior of the solution to a degenearing mixed Neumann-Robin problem in Ω(ε) as the size ε of the small hole tends to 0. In addition to the geometric degeneracy of the problem, the nonlinear ε-dependent Robin condition may degenerate into a Neumann condition for ε = 0 and the Robin datum may diverge to infinity. Our goal was to prove the existence of solutions for ε small and positive and to study the corresponding asymptotic behavior as ε→ 0. Instead of the more common methods of Asymptotic Analysis dealing with asymptotic expansions, here we have employed the Functional Analytic Approach proposed by Lanza de Cristoforis. Such method has the advantages to be applicable to nonlinear boundary conditions as in the present paper and to provide rigorous justifications to power series expansions. Moreover, such approach is quite versatile: it has been applied to elliptic systems of partial differential equations (see Dalla Riva and Lanza de Cristoforis [5, 6]) and currently some preliminary results have been obtained also in order to consider parabolic equations (see Dalla Riva and Luzzini [10] and Luzzini [23]). Acknowledgements. The authors acknowledge the support from EU through the H2020-MSCA-RISE-2020 project EffectFact, Grant agreement ID: 101008140. The authors thank Dr. Luigi Provenzano for valuable discussions on Steklov eigenvalues in relation to the toy problem of Section 2. P. Musolino also acknowledges the support of the SPIN Project “DOMain perturbation problems and INteractions Of scales - DOMINO” of the Ca’ Foscari University of Venice. Part of the work was done while P. Musolino was visiting M. Dutko at Rockfield Software Limited. P. Musolino wishes to thank M. Dutko and Rockfield Software Limited for the kind hospitality. P. Musolino is a member of the Gruppo Nazionale per l’Analisi Matem- atica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 19 Alta Matematica (INdAM). G. Mishuris acknowledges also Ser Cymru Future Gen- eration Industrial Fellowship number AU224 – 80761. M. Dutko also acknowledges the Royal Academy of Engineering for the Industrial Fellowship. References [1] H. Ammari, H. Kang; Polarization and moment tensors, volume 162 of Applied Mathematical Sciences, Springer, New York, 2007 [2] H. Ammari, J.-C. Nédélec; Generalized impedance boundary conditions for the Maxwell equa- tions as singular perturbations problems, Comm. Partial Differential Equations, 24 (5-6) (1999), 821–849. [3] R. Böhme, F. Tomi; Zur Struktur der Lösungsmenge des Plateauproblems, Math. Z., 133 (1973), 1–29. [4] M. Costabel, M. Dauge; A singularly perturbed mixed boundary value problem, Comm. Partial Differential Equations, 21(11-12) (1996), 1919–1949. [5] M. Dalla Riva, 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(8-10) (2010), 771–794. [6] M. Dalla Riva, M. Lanza de Cristoforis; A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach, Analysis (Munich), 30(1) (2010), 1, 67–92. [7] M. Dalla Riva, M. Lanza de Cristoforis; Hypersingularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach, Eurasian Math. J., 1(2) (2010), 31–58. [8] M. Dalla Riva, 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(3) (2011), 811–833. [9] M. Dalla Riva, M. Lanza de Cristoforis, P. Musolino; Singularly perturbed boundary value problems. A functional analytic approach. Springer, Cham, 2021. [10] M. Dalla Riva, P. Luzzini; Dependence of the layer heat potentials upon support perturbations, Differential Integral Equations 36(11-12) (2023), 971–1003. [11] K. Deimling; Nonlinear functional analysis. Springer-Verlag, Berlin, 1985. [12] Elfen Glass Design, https://www.elfen-glassdesign.com/ Accessed on December 7, 2022. [13] Elfen Welbore https://www.rockfieldglobal.com/software/elfen-wellbore/ Accessed on De- cember 7, 2022. [14] D. Gilbarg, NS. Trudinger; Elliptic partial differential equations of second order Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin; second ed, 1983 [15] M. Grossi, P. Luo; Critical points of positive solutions of nonlinear elliptic equations: mul- tiplicity, location, and non-degeneracy, Indiana Univ. Math. J. 72 (2023), 821-871. [16] D. Henry; Topics in Nonlinear Analysis. Trabalho de Matemática, 1982. [17] A. M. Il’in; Matching of asymptotic expansions of solutions of boundary value problems, Translations of Mathematical Monographs 102, American Mathematical Society, Providence, 1992. [18] A. Kirsch; The Robin problem for the Helmholtz equation as a singular perturbation problem, Numer. Funct. Anal. Optim., 8(1-2) (1985), 1–20. [19] 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(1) (2002), 1–27. [20] 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(10-11) (2007), 945–977. [21] M. Lanza de Cristoforis, P. Musolino; A real analyticity result for a nonlinear integral oper- ator, J. Integral Equations Appl., 25(1) (2013), 21–46. [22] M. Lanza de Cristoforis, L. Rossi; Real analytic dependence of simple and double layer poten- tials upon perturbation of the support and of the density, J. Integral Equations Appl. 16(2) (2004), 137–174. 20 P. MUSOLINO, M. DUTKO, G. MISHURIS EJDE-2023/57 [23] P. Luzzini; A mapping property of the heat volume potential, Boll. Unione Mat. Ital., to appear. [24] V. G. Maz’ya, A. B. Movchan, M. J. Nieves; Green’s kernels and meso-scale approximations in perforated domains, Lecture Notes in Mathematics, vol. 2077. Springer, Heidelberg, 2013. [25] V.G. Maz’ya, A. B. Movchan, M. J. Nieves; Mesoscale approximations for solutions of the Dirichlet problem in a perforated elastic body, J. Math. Sci. (N.Y.) 202(2) (2014), Problems in mathematical analysis. No. 76 (Russian): 215–244. [26] V. G. Maz’ya, A. B. Movchan, M. J. Nieves; Mesoscale models and approximate solutions for solids containing clouds of voids, Multiscale Model. Simul. 14(1) (2016), 138–172. [27] V. G. Maz’ya, A. B. Movchan, M. J. Nieves; Eigenvalue problem in a solid with many inclusions: asymptotic analysis, Multiscale Model. Simul. 15(2) (2017), 1003–1047. [28] V. G. Maz’ya, A. B. Movchan, M. J. Nieves; On mesoscale approximations for vibrations of membranes with lower-dimensional clusters of inertial inclusions, Algebra i Analiz, 32(3): 219–237 (2020); reprinted in St. Petersburg Math. J. 32(3) (2021), 551–564 [29] V. Maz’ya, S. Nazarov, B. Plamenevskij; Asymptotic theory of elliptic boundary value prob- lems in singularly perturbed domains. Vol. I Operator Theory: Advances and Applications, vol. 111, Birkhäuser Verlag, Basel, 2000. Translated from the German by Georg Heinig and Christian Posthoff. [30] V. Maz’ya, S. Nazarov, B. Plamenevskij; Asymptotic theory of elliptic boundary value prob- lems in singularly perturbed domains. Vol. II Operator Theory: Advances and Applications, vol. 112, Birkhäuser Verlag, Basel, 2000. Translated from the German by Plamenevskij. [31] C. Miranda; Sulle proprietà di regolarità di certe trasformazioni integrali, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I, (8) 7 (1965), 303–336. [32] G. Mishuris; Imperfect transmission conditions for a thin weakly compressible interface. 2D problems, Arch. Mech. (Arch. Mech. Stos.), 56(2) (2004), 103–115. [33] G. Mishuris, W. Miszuris, A. Öchsner; Evaluation of Transmission Conditions for Thin Reactive Heat-Conducting Interphases, In Defect and Diffusion Forum, vol. 273: 394–399, Trans Tech Publications, 2008. [34] G. Mishuris, W. Miszuris, A. Öchsner; Transmission Conditions for Thin Reactive Heat- Conducting Interphases: General Case, Defect and Diffusion Forum, vol. 283: 521–526,T rans Tech Publications, 2009 [35] P. Musolino, G. Mishuris; A nonlinear problem for the Laplace equation with a degenerating Robin condition, Math. Methods Appl. Sci., 41(13) (2018), 5211–5229. [36] P. Musolino, G. Mishuris; Interaction of scales for a singularly perturbed degenerating non- linear Robin problem, Philos. Trans. Roy. Soc. A, 380(2236) (2022), 20220159. [37] M. J. Nieves, A. B. Movchan; Asymptotic analysis of in-plane dynamic problems for elastic media with rigid clusters of small inclusions, Philos. Trans. Roy. Soc. A, 380(2237) (2022), 20210392 [38] M. J. Nieves; Asymptotic analysis of solutions to transmission problems in solids with many inclusions, SIAM J. Appl. Math., 77(4) (2017), 1417–1443. [39] A. A. Novotny, J. Soko lowski; Topological derivatives in shape optimization, Interaction of Mechanics and Mathematics, Springer, Heidelberg, 2013 [40] K. Schmidt, R. Hiptmair; Asymptotic expansion techniques for singularly perturbed boundary integral equations, Numer. Math., 137(2) (2017), 397–415. [41] T. Valent; Boundary value problems of finite elasticity Springer Tracts in Natural Philoso- phy, vol. 31. Springer-Verlag, New York, 1999. Local theorems on existence, uniqueness, and analytic dependence on data. [42] M. Th. van Genuchten, W. J. Alves; Analytical Solutions of the One -Dimensional Convective-Dispersive Solute Transport Equation, Technical Bulletin - United States De- partment of Agriculture, 1661 (1982). [43] W. L. Wendland, E. Stephan, G. C. Hsiao; On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci., 1(3) (1979), 265–321. Paolo Musolino Dipartimento di Scienze Molecolari e Nanosistemi, Università Ca’ Foscari Venezia, via Torino 155, 30172 Venezia Mestre, Italy Email address: paolo.musolino@unive.it EJDE-2023/57 ASYMPTOTIC ANALYSIS OF PERTURBED ROBIN PROBLEMS 21 Martin Dutko Rockfield Software Limited, King’s Road, Ethos Building, Swansea, SA1 8PH, Wales UK Email address: martin.dutko@rockfieldglobal.com Gennady Mishuris Department of Mathematics, Aberystwyth University, Ceredigion, Aberystwyth, SY23 3BZ Wales, UK Email address: ggm@aber.ac.uk 1. Introduction 2. A toy problem 3. Integral equation formulation of the boundary value problem 4. Analytic representation formulas for the solution of the boundary value problem 5. Remarks on the linear case 6. Conclusions Acknowledgements References