A nonlinear problem for the Laplace equation with a degenerating Robin condition

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution.


INTRODUCTION
In this paper, we study the asymptotic behavior of the solutions of a boundary value problem for the Laplace equation with a (nonlinear) Robin boundary condition, which degenerates into a Neumann condition.
Boundary value problems with perturbed Robin or mixed conditions have been investigated by several authors. For example, Wendland et al 1 considered a family of Poincaré problems approximating a mixed boundary value problem for the Laplace equation in the plane. Kirsch 2 studied the convergence of the solution of the Helmholtz equation with boundary condition of the type − u +u = g to the solution with Dirichlet condition u = g as → 0. Costabel and Dauge 3 studied a mixed Neumann-Robin problem for the Laplace operator, where the Robin condition contains a parameter so that it tends to a Dirichlet condition as → 0. An extension to nonlinear equation has been considered, for example, in Berestycki and Wei. 4 Degenerating nonlinear Robin conditions in the frame of homogenization problems have been studied by Gómez et al. 5 Singularly perturbed boundary conditions for the Maxwell equations have been analyzed, for example, in Ammari and Nédélec. 6 Moreover, Schmidt and Hiptmair 7 have exploited integral equation methods for singularly perturbed boundary conditions in the frame of transmission problems. Furthermore, an approach based on potential theory to prove the solvability of a small nonlinear perturbation of a homogeneous linear transmission problem can be found in Dalla Riva and Mishuris. 8 Concerning existence and uniqueness results for boundary value problems with nonlinear Robin conditions, we also mention, eg, Donato et al. 9 We note that the transmission problem for a composite domain with imperfect (nonnatural) conditions along the joint boundary is, in fact, a generalization of the classical Robin problem. Such transmission conditions frequently appear in practical applications for various nonlinear multiphysics problems (eg, Mishuris et al 10,11 and Mishuris 12 ). Moreover, the imperfect transmission conditions allow one to perform numerical analysis of practical problems with thin interphases at low cost with sufficient accuracy (see Mishuris and Öchsner, 13 Mishuris et al, 14 and Sonato et al 15 ).
In this paper, instead, we are interested in the case where the Robin condition degenerates into a Neumann condition.
To introduce the problem, we first define the geometric setting. We fix once for all a natural number n ∈ N∖{0, 1}.
Then, we consider ∈]0, 1[ and two subsets Ω i , Ω o of R n satisfying the following assumption: Ω i andΩ o are bounded open connected subsets ofR n of classC 1, such that Ω i ⊆ Ω o and that R n ∖Ω i andR n ∖Ω o areconnected.
For the definition of sets and functions of the Schauder class C k, (k ∈ N), we refer, eg, to Gilbarg and Trudinger. 16 The letter "i" stands for "inner" and the letter "o" stands for "outer." The symbol "·" denotes the closure. Then, we introduce the domain Ω by setting We note that the boundary Ω of Ω consists of the two connected components Ω o and Ω i . Therefore, we can identify, for example, C 0, ( Ω) with the product C 0, ( Ω o ) × C 0, ( Ω i ). To define the boundary data, we fix two functions Then, we take 0 > 0 and a family {F } ∈]0, 0 [ of functions from R to R. Next, for each ∈]0, 0 [, we want to consider a nonlinear boundary value problem for the Laplace operator. Namely, we consider a Neumann condition on Ω o and a nonlinear Robin condition on Ω i . Thus, for each ∈]0, 0 [, we consider the following boundary value problem: where Ω o and Ω i denote the outward unit normal to Ω o and to Ω i , respectively. As a first step, under suitable assumptions, in this paper, we show that for each positive and small enough, problem (1) has a solution, which we denote by u( , ·). Then, we are interested in studying the behavior of u( , ·) as → 0, and thus, we pose the following questions.
(1) Let x be a fixed point in Ω. What can be said of the map  → u( , x) when is close to 0 and positive? (2) What can be said of the map  → ∫ Ω |∇u( , x)| 2 dx when is close to 0 and positive?
We also note that if in correspondence of the limiting value = 0, we omit the term in (1), then we obtain the Neumann problem On the other hand, by the divergence theorem and classical existence results for the Neumann problem, problem (2) has (at least) a solution if and only if This means, in particular, that if (3) does not hold, then u( , ·) cannot converge to a solution of problem (2) as → 0.
In contrast with asymptotic expansion methods, in this paper, we answer the questions in (1), (2) by representing the maps of (1), (2) in terms of real analytic maps in Banach spaces and in terms of known functions of (for the definition and properties of real analytic maps, we refer to Deimling 17,p. 150 ). We observe that if, for example, we know that the function in (1) equals for > 0 a real analytic function defined in a whole neighborhood of = 0, then we know that such a map can be expanded in power series for small.
Such an approach has been proposed by Lanza de Cristoforis 18 for the analysis of singularly perturbed problems in perforated domain as an alternative to asymptotic expansion methods (cf, eg, Maz'ya et al 19 and Maz'ya et al 20,21 ). In particular, it has been exploited to analyze singularly perturbed (linear and nonlinear) Robin and mixed problems in domains with small holes (cf, eg, Lanza de Cristoforis 22 and Dalla Riva and Musolino 23 for the Laplace equation and Dalla Riva and Lanza de Cristoforis 24,25 for the Lamé equations).
The paper is organized as follows. In Section 2, we consider some model problems in an annular domain where we can explicitly construct the solutions and discuss the behavior as tends to 0. In Section 3, we formulate our problem in terms of integral equations. In Section 4, we prove our main result, which answers our questions (1), (2) above, and in Section 5 we discuss a local uniqueness property of the family of solutions. Finally, in Section 6, we make some comments on the linear case and compute the power series expansion of the solution.

MODEL PROBLEMS
To illustrate some aspects of the problem under investigation, in this section, we consider the set Ω ≡ B n (0, 1)∖B n (0, 1∕2), ie, we take Ω o ≡ B n (0, 1) and Ω i ≡ B n (0, 1∕2), where, for r > 0, the symbol B n (0, r) denotes the open ball in R n of center 0 and radius r.

A linear problem
We begin with a linear problem and to do so, we take a, b ∈ R. Then, for each ∈]0, + ∞[, we consider the problem As is well known, for each ∈]0, + ∞[, problem (4) has a unique solution in C 1, (Ω), and we denote it by u . On the other hand, if instead we put = 0 in (4) we obtain The solvability of problem (5) is subject to a compatibility condition on the Neumann data on B n (0, 1) and on B n (0, 1∕2). More precisely, problem (5) has a solution if and only if ie, if and only if where s n denotes the (n − 1)-dimensional measure of B n (0, 1). Condition (6) can be rewritten as follows: In particular, if a = b 2 n−1 then, the Neumann problem (5) has a 1-dimensional space of solutions; if instead a ≠ b 2 n−1 , problem (5) does not have any solution.
This implies that in general the solution u of problem (4) cannot converge to a solution of (5) as → 0, if the compatibility condition (7) does not hold. Therefore, we wish to understand the behavior of u as → 0, and we do so by constructing explicitly u .
To construct the solution u , we consider separately case n = 2 and n ≥ 3. If n = 2, we look for the function u in the form with A and B to be set so that the boundary conditions of problem (4) are satisfied. We first note that and that accordingly which implies that we must have in order to fulfill the Neumann condition on B n (0, 1). On the other hand, as far as the Robin condition on B 2 (0, 1∕2) is concerned, we must find B such that Then, a straightforward computation implies that we must have As a consequence, if n = 2, we have Then, we turn to consider the case of dimension n ≥ 3, and we look for a solution of problem (4) in the form with A and B to be set so that the boundary conditions of problem (4) are satisfied. By arguing as above, one deduces that and thus, Thus, by looking at (8) and (9), we note that if condition (7) does not hold, then, Comparing (8) and (9) one can write the solutions in a uniform manner: where ∀x ∈Ω, In particular, we note that u (0) is the unique solution of (5) such that On the other hand, if (7) holds, we have u (1) ≡ 0 and for all ∈]0, + ∞[, and u is also a solution to problem (5).

A nonlinear problem
In this section, we analyze a nonlinear problem, and for the sake of simplicity, we confine to the case of dimension n = 2. For each ∈]0, + ∞[, we consider the problem Now, we note that we can collect in the right hand side of the third equation in (11), and thus, we can write the Robin condition as follows: If for each ∈]0, + ∞[, we introduce the function we can rewrite problem (11) as follows: Then again, we look for a solution u in the form with A and B to be set so that the boundary conditions of problem (12) are satisfied.
As we have seen, to ensure the validity of the Neumann condition on B 2 (0, 1), we must have On the other hand, in order to satisfy the Robin condition, we have to find B such that Motivated by the linear case, we find it convenient to replace B byB ∕ + a log 2. In other words, we look for a solution u in the form Then, we note that if we setF As a consequence, we can rewrite Equation 13 as follows: For generalF, under suitable assumptions, one can try to resolve Equation 15 by means of the implicit function theorem.
On the other hand, for our specific case, for each ∈]0, + ∞[, one has that the solutions in C of equation Thus, if we look for solutionsB ∈ R of Equation 15 for positive and close to 0, we may have 1, 2, or no solutions to (15) depending on the sign of Therefore, for small and positive, we may have 1, 2, or no solutions to the nonlinear problem (11). In particular, a crucial role for the solvability of problem (11) is played by the functionF, which ensures the validity of Equation 14.

A family of nonlinear problems
To play with the structure of the nonlinear boundary condition, for each ∈]0, + ∞[, we consider the family of problems where c ∈ R and 1 , 2 ∈ N. Note that such type of boundary conditions is crucially important for practical applications. For example, in metallurgy and metal forming processes, the typical boundary condition involves 2 = 4 where the respective term corresponds to the heat exchange due to the radiation at high temperature (see Golitsyna, 26 Letavin and Mishuris, 27 and Letavin and Shestakov 28 ). Now, we note that we can rewrite the Robin condition as follows: As above, for each ∈]0, + ∞[, we introduce the function Then, we can rewrite problem (16) as follows: Again, we look for a solution u in the form with A andB to be set so that the boundary conditions of problem (17) are satisfied.
As we have seen, we must have and, in order to satisfy the Robin condition, we have to findB such that Then, we note that if we setF Since we want to pass to the limit inF( , 1 − 2 ) as → 0, we find it convenient to assume that As a consequence, we rewrite Equation 18 as follows: We try to resolve Equation 19 around = 0 by means of the implicit function theorem. We treat separately the case 1 = 2 and the case 1 > 2 . If 1 > 2 , then, there exists a uniqueB 0 such that Then, by applying the implicit function theorem around the pair (2a − b, 0), one can prove that there exist a small , we turn to consider the case 1 = 2 , and we note that As a consequence, there are 1 + 1 complex solutions to the equation Then, if we denote by {B } k =1 the set of (distinct) real solutions to Equation 20 for each of them, we can construct the corresponding function, and thus, we can define a family of solutions {u j, } ∈]0, + ∞[ to problem (16), by setting Note that this can be presented in the form: thus, the nonuniqueness is related to the second term of this representation only. Moreover, it makes sense also to underline that the first term in the solutions for the linear (10) and nonlinear (21) problems coincides.

AN INTEGRAL EQUATION FORMULATION OF THE BOUNDARY VALUE PROBLEM
To analyze problem (1) for close to 0, we exploit classical potential theory, which allows to obtain an integral equation formulation of (1). To do so, we need to introduce some notation. Let S n be the function from R n ∖{0} to R defined by S n is well known to be a fundamental solution of the Laplace operator. We now introduce the single layer potential.
where d denotes the area element of a manifold imbedded in R n . As is well known, if ∈ C 0 ( Ω), where Ω denotes the outward unit normal to Ω. If ∈ C 0, ( Ω), the function w * [ Ω, ] belongs to C 0, ( Ω), and we have Then, we have the technical Lemma 1 below on the representation of harmonic functions as the sum of a single layer potential with a density with zero integral mean and a constant. Therefore, we find it convenient to set The proof of Lemma 1 can be deduced by classical potential theory (cf Folland 29, ch. 3 ).
By exploiting Lemma 1, we can establish a correspondence between the solutions of problem (1) and those of a (nonlinear) system of integral equations.
to the set of those functions u ∈ C 1, (Ω) that solve problem (1) is a bijection.
Proof. If ( , ) ∈ C 0, ( Ω) × R, then we know that v + [ Ω, ] + belongs to C 1, (Ω) and is harmonic in Ω. Moreover, if ( , ) satisfies system (22), then the jump formulas for the normal derivative of the single layer potential imply the validity of the boundary condition in problem (1). Hence, the function in (23) solves problem (1). Conversely, if u ∈ C 1, (Ω) satisfies problem (1), then the representation Lemma 1 for harmonic functions in terms of single layer potentials plus constants ensures that there exists a unique pair ( , ) ∈ C 0, ( Ω) 0 × R such that u = v + [ Ω, ] + . Then, the jump formulas for the normal derivative of a single layer potential and the boundary condition in (1) imply that the system of integral equations of (22) is satisfied. Hence, the map of the statement is a bijection. Now that the correspondence between the solutions of boundary value problem (1) and those of the system of integral equations (22) is established, we wish to study the behavior of the solutions to system (22) as → 0. Then, we note that we can write ) .
Therefore, to analyze the second equation in (22) for small, we need to make some other assumptions on the structure of the family of functions Thus, under the additional assumption (24), if we let tend to 0 in (22), we obtain the following limiting system of integral equations: Then, as a preliminary step in the analysis of the system of integral equations (22) for close to 0, in the following lemma, we study the limiting system (25).
In view of Proposition 1 and under assumption (24), in order to study the solutions of (22), we find it convenient to introduce the map Λ In the following proposition, we investigate the solutions of the system of integral equations (22), by applying the implicit function theorem to Λ, under suitable assumptions on the partial derivativeF(̃, 0 ) of the function ( , )  → F( , ) with respect to the variable computed at the point (̃, 0 ). Proof. We first note that by classical potential theory (cf Miranda 30 and Lanza de Cristoforis and Rossi 31, thm. 3.1 ), by assumption (24), and by analyticity results for the composition operator (cf Böhme and Tomi, 32, p. 10 Henry, 33, p. 29 and Valent 34, Thm. 5.2, p. 44 ), we conclude that Λ is real analytic. Then, we note that the partial differential ( , ) Λ[0, 0 ,̃,̃] of Λ at (0, 0 ,̃,̃) with respect to the variable ( , ) is delivered by

A FUNCTIONAL ANALYTIC REPRESENTATION THEOREM FOR THE FAMILY OF SOLUTIONS
In the following theorem, we exploit the analyticity result of Proposition 2 concerning the solutions of the system of integral equations (22) in order to prove representation formulas for u( , ·) and its energy integral in terms of real analytic maps and thus to answer to questions (1), (2) of the Introduction. and (2) There exists a real analytic map E from ] − 2 , 2 [× to R such that for all ∈]0, 2 [. Moreover, whereũ is any solution of the Neumann problem (29).
Proof. We first prove statement (1). We set for all ( , ) ∈]− 2 , 2 [× . Then, by Proposition 2 and by classical mapping properties of layer potentials (cf Miranda 30 and Lanza de Cristoforis and Rossi 31, thm. 3.1 ), we conclude that U is real analytic. Sincẽ As a consequence, v + [ Ω,̃] solves problem (29). Then, we deduce the validity of statement (1) (see also Proposition 2). We now consider statement (2). By the divergence theorem and standard properties of harmonic functions and their normal derivatives, we have for all ∈]0, 2 [. Thus, we find natural to set By Proposition 2, by mapping properties of layer potentials, and by standard calculus in Schauder spaces, we deduce the real analyticity of E from ] − 2 , 2 [× to R. Since and v + [ Ω,̃] is a solution of problem (29), we deduce the validity of statement (2).
Remark 1. We observe that Theorem 1 implies that the quantities in the left-hand sides of (28) and of (30) can be represented as convergent power series of ( , ( ) − 0 ).

LOCAL UNIQUENESS OF THE FAMILY OF SOLUTIONS
We now show by means of the following theorem that the family {u( , ·)} ∈]0, 2 [ is locally essentially unique (cf Lanza de Cristoforis 22, thm. 4.1 (iii) ).
then, there exists 0 ∈ N such that u j (·) = u(d j , ·) for all j ≥ j 0 .

REMARKS ON THE LINEAR CASE
In this section, we wish to make further considerations on the linear case. In particular, we plan to compute asymptotic expansions of the solutions as the parameter tends to 0. We first note that the results of Section 4 apply to the linear case. In particular, in case problem (1) reduces to the following linear problem: For each ∈]0, + ∞[, we know that problem (37) has a unique solution in C 1, (Ω), and we denote it by u[ ]. Clearly, and thus, we can take, for example, In particular, Therefore, the results of Sections 3 and 4 apply to the present case. More precisely, by simplifying the arguments of Propositions 1 and 2, we deduce the validity of the following proposition. Moreover, and (2) There exists a real analytic map whereũ is any solution of the Neumann problem (39).

Asymptotic expansion of u[ ]
By Theorem 3 (1), we know that there exist a sequence of functions {u #,k } k∈N ⊆ C 1, (Ω) and a sequence of real numbers where the series are uniformly convergent for in a neighborhood of 0. As for the model problem (10), we note that we can rewrite equation (40) in the form where in this case in general u (1) depends on .
To construct the sequences {u #,k } k∈N ⊆ C 1, (Ω) and { #,k } k∈N , we wish to exploit the integral equation formulation of problem (37) and the approach of Dalla Riva et al. 36 Now, we observe that the real analyticity result of Proposition 3 implies that there exists 1 ∈]0, 0 [ small enough such that we can expand M # [ ] and # [ ] into power series of , ie, The plan is to obtain such equations by deriving with respect to equality (38), which then leads to Then, as Proposition 4 below shows, by taking = 0 in (42), we will obtain integral equations identifying and where the two series converge uniformly for ∈] − 1 , 1 [. Moreover, the following statements hold.
Finally, by Propositions 3 and 4, Theorem 4 and standard calculus in Banach spaces, one deduces the validity of the following.