Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem

Nonlinear Analysis 191 (2020) 111645

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Local uniqueness of the solutions for a singularly perturbed nonlinear nonautonomous transmission problem Matteo Dalla Riva a ,∗, Riccardo Molinarolo b , Paolo Musolino c a

Department of Mathematics, The University of Tulsa, 800 South Tucker Drive, Tulsa, OK 74104, USA Department of Mathematics, Aberystwyth University, Ceredigion SY23 3BZ, Wales, United Kingdom Dipartimento di Matematica ‘Tullio Levi-Civita’, Università degli Studi di Padova, Via Trieste 63, Padova 35121, Italy b c

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info

Article history: Received 14 December 2018 Accepted 29 September 2019 Communicated by Vicentiu D. Radulescu MSC: 35J25 31B10 35J65 35B25 35A02

abstract We consider the Laplace equation in a domain of Rn , n ≥ 3, with a small inclusion of size ϵ. On the boundary of the inclusion we define a nonlinear nonautonomous transmission condition. For ϵ small enough one can prove that the problem has solutions. In this paper, we study the local uniqueness of such solutions. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Nonlinear nonautonomous transmission problem Local uniqueness of the solutions Singularly perturbed perforated domain

1. Introduction We study local uniqueness properties of the solutions of a nonlinear nonautonomous transmission problem for the Laplace equation in the pair of sets consisting of a perforated domain and a small inclusion. We begin by presenting the geometric framework of our problem. We fix once for all a natural number n≥3 that will be the dimension of the space Rn we are going to work in and a parameter α ∈ ]0, 1[ ∗ Corresponding author. E-mail addresses: [email protected] (M. Dalla Riva), [email protected] (R. Molinarolo), [email protected] (P. Musolino).

https://doi.org/10.1016/j.na.2019.111645 0362-546X/© 2019 Elsevier Ltd. All rights reserved.

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M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

which we use to define the regularity of our sets and functions. We remark that the case of dimension n = 2 requires specific techniques and it is not treated in this paper (the analysis for n = 3 and for n ≥ 3 is instead very similar). Then, we introduce two sets Ω o and Ω i that satisfy the following conditions: Ω o , Ω i are bounded open connected subsets of Rn of class C 1,α , their exteriors Rn \ Ω o and Rn \ Ω i are connected, and the origin 0 of Rn belongs both to Ω o and to Ω i . Here the superscript “o” stands for “outer domain” whereas the superscript “i” stands for “inner domain”. We take ϵ0 ≡ sup{θ ∈ ]0, +∞[ : ϵΩ i ⊆ Ω o , ∀ϵ ∈ ] − θ, θ[}, and we define the perforated domain Ω (ϵ) by setting Ω (ϵ) ≡ Ω o \ ϵΩ i for all ϵ ∈ ] − ϵ0 , ϵ0 [. Then we fix three functions F : ] − ϵ0 , ϵ0 [ × ∂Ω i × R → R , G : ] − ϵ0 , ϵ0 [ × ∂Ω i × R → R , o

f ∈C

1,α

(1)

o

(∂Ω )

and, for ϵ ∈ ]0, ϵ0 [, we consider the following nonlinear nonautonomous transmission problem in the perforated domain Ω (ϵ) for a pair of functions (uo , ui ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ): ⎧ ⎪ ∆uo = 0 in Ω (ϵ), ⎪ ⎪ ⎪ i ⎪ ⎪ ∆u = 0 in ϵΩ i , ⎨ (2) uo (x) = f o (x) ∀x ∈ ∂Ω o , ⎪ ( x i ) ⎪ o i ⎪ u (x) = F ϵ, , u (x) ∀x ∈ ϵ∂Ω , ⎪ ϵ ⎪ ) ( ⎪ ⎩ν o · ∇u (x) − νϵΩ i · ∇ui (x) = G ϵ, xϵ , ui (x) ∀x ∈ ϵ∂Ω i . i ϵΩ Here νϵΩ i denotes the outer exterior normal to ϵΩ i . Boundary value problems like (2) arise in the mathematical models for the heat conduction in (nonlinear) ¨ composite materials, as, for example, in Mishuris, Miszuris and Ochsner [18,19]. In such cases, the functions uo and ui represent the temperature distribution in Ω (ϵ) and in the inclusion ϵΩ i , respectively. The third condition in (2) means that we are prescribing the temperature distribution on the exterior boundary ∂Ω o . The fourth condition says that on the interface ϵ∂Ω i the temperature distribution uo depends nonlinearly on the size of the inclusion, on the position on the interface, and on the temperature distribution ui . The fifth conditions, instead, says that the jump of the heat flux on the interface depends nonlinearly on the size of the inclusion, on the position on the interface, and on the temperature distribution ui . In literature, existence and uniqueness of solutions of nonlinear boundary value problems have been largely investigated by means of variational techniques (see, e.g., the monographs of Neˇcas [21] and of Roub´ıˇcek [22] and the references therein). On the other hand, boundary value problems in singularly perturbed domains are usually studied by expansion methods of asymptotic analysis, such as the methods of matching inner and outer expansions (cf., e.g., Il’in [15,16]) and the multiscale expansion method (as in Maz’ya, Nazarov, and Plamenenvskii [17], see also Iguernane et al. [14] in connection with nonlinear problems). In this paper we do not use variational techniques and neither we resort to asymptotic expansion methods. Instead, we adopt the functional analytic approach proposed by Lanza de Cristoforis (cf., e.g., [3,5]). The key strategy of such approach is the transformation of the perturbed boundary value problem into a functional

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equation that can be studied by means of certain theorems of functional analysis, for example by the implicit function theorem or by the Schauder fixed-point theorem. Typically, this transformation is achieved with an integral representation of the solution, a suitable rescaling of the corresponding integral equations, and an analysis based on results of potential theory. This approach has proven to be effective when dealing with nonlinear conditions on the boundary of small holes or inclusions. For example, it has been used in the papers [4] and [6] of Lanza de Cristoforis to study a nonlinear Robin and a nonlinear transmission problem, respectively, in the paper [8] with Lanza de Cristoforis to analyze a nonlinear traction problem for an elastic body, in [9] with Mishuris to prove the existence of solution for (2) in the case of a “big” inclusion (that is, for ϵ > 0 fixed), and in [20] to show the existence of solution of (2) in the case of “small” inclusion (that is, for ϵ > 0 that tends to 0). In particular, in [20] we have proven that, under suitable assumptions on the functions F and G, there exists a family of functions {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ , with ϵ′ ∈ ]0, ϵ0 [, such that each pair (uoϵ , uiϵ ) is a solution of (2) for the corresponding value of ϵ. Moreover, the dependence of the functions uoϵ and uiϵ upon the parameter ϵ can be described in terms of real analytic maps of ϵ. The aim of this paper is to show that each of such solutions (uoϵ , uiϵ ) is locally unique. In other words, we shall verify that, for ϵ > 0 smaller than a certain ϵ∗ ∈ ]0, ϵ0 [, any solution (v o , v i ) of problem (2) that is “close enough” to the pair (uoϵ , uiϵ ) has to coincide with (uoϵ , uiϵ ). We will see that the “distance” from the solution (uoϵ , uiϵ ) can be measured solely in terms of the C 1,α norm of the trace of the rescaled function v i (ϵ·) on ∂Ω i . More precisely, we will prove that there is δ ∗ > 0 such that, if ϵ ∈ ]0, ϵ∗ ], (v o , v i ) is a solution of (2), and   i v (ϵ·) − uiϵ (ϵ·) 1,α i < ϵδ ∗ , (3) C (∂Ω ) then (v o , v i ) = (uoϵ , uiϵ ) (cf. Theorem 5.6). We note that in [9] it has been shown that for a “big” inclusion (that is, with ϵ > 0 fixed) problem (2) may have solutions that are not locally unique. Such a different result reflects the fact that the solutions of [9] are obtained by the Schauder fixed-point theorem, whereas the solutions of the present paper are obtained by means of the implicit function theorem. We will not provide in this paper an estimate for the values of ϵ∗ and δ ∗ . In principle, they could be obtained studying the norm of certain integral operators that we employ in our proofs. We observe that, in specific applications, for example to the heat conduction in composite materials, it might be important to understand if ϵ∗ and δ ∗ are big enough for the model that one adopts. In particular, for very small ϵ∗ , that correspond to very small inclusions, and very small δ ∗ , that corresponds to very small differences of temperature, one may have to deal with different physical models. We also observe that uniqueness results are not new in the applications of the functional analytic approach to nonlinear boundary value problems (see, e.g., the above mentioned papers [4,6,8]). However, the results so far presented concern the uniqueness of the entire family of solutions rather than the uniqueness of a single solution for ϵ > 0 fixed. For our specific problem (2), a uniqueness result for the family {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ would consist in proving that if {(vϵo , vϵi )}ϵ∈]0,ϵ′ [ is another family of solutions which satisfies a certain limiting condition, for example that   lim ϵ−1 vϵi (ϵ·) − uiϵ (ϵ·)C 1,α (∂Ω i ) = 0, ϵ→0

then (vϵo , vϵi ) = (uoϵ , uiϵ ) for ϵ small enough. One can verify that the local uniqueness of a single solution under condition (3) implies the uniqueness of the family of solutions {(uoϵ , uiϵ )}ϵ∈]0,ϵ [ in the sense described here above (cf. Corollary 5.7). From this 0

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point of view, we can say that the uniqueness result presented in this paper strengthen the uniqueness result for families which is typically obtained in the application of the functional analytic approach. The paper is organized as follows. In Section 2 we define some of the symbols used later on. Section 3 is a section of preliminaries, where we introduce some classical results of potential theory that we need. In Section 4 we recall some results of [20] concerning the existence of a family {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ of solutions of problem (2). In Section 5 we state and prove our main Theorem 5.6. The section consists of three subsections. In the first one we prove Theorem 5.2, which is a weaker version of Theorem 5.6 and follows directly from the Implicit Function Theorem argument used to obtain the family {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ . The statement of Theorem 5.2 is similar to that of Theorem 5.6, but the assumptions are much stronger. In particular, together with condition (3), we have to require other two conditions, namely that ∥v o − uoϵ ∥C 1,α (∂Ω o ) < ϵδ ∗

and

∥v o (ϵ·) − uoϵ (ϵ·)∥C 1,α (∂Ω i ) < ϵδ ∗ ,

(4)

in order to prove that (v o , v i ) = (uoϵ , uiϵ ). In our main Theorem 5.6 we will see that the two conditions in (4) can be dropped and (3) is sufficient. The proof of Theorem 5.6 is presented in Section 5.2 where we first show some results on real analytic composition operators in Schauder spaces, then we turn to introduce certain auxiliary maps N and S, and finally we will be ready to state and prove our main theorem. In the last Section 5.3, we see that the uniqueness of the family {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ in the sense described here above can be obtained as a corollary of Theorem 5.6. At the end of the paper we have included an Appendix where we present some (classical) results on the product and composition in Schauder spaces. 2. Notation We denote the norm of a real normed space X by ∥ · ∥X . We denote by IX the identity operator from X to itself and we omit the subscript X where no ambiguity can occur. For x ∈ X and R > 0, we denote by BX (x, R) ≡ {y ∈ X : ∥y − x∥X < R} the ball of radius R centered at the point x. If X = Rd , d ∈ N \ {0, 1}, we simply write Bd (x, R). If X and Y are normed spaces we endow the product space X × Y with the norm defined by ∥(x, y)∥X×Y = ∥x∥X + ∥y∥Y for all (x, y) ∈ X × Y , while we use the Euclidean norm for Rd , d ∈ N \ {0, 1}. For x ∈ Rd , xj denotes the jth coordinate of x, |x| denotes the Euclidean modulus of x in Rd . We denote by L(X, Y ) the Banach space of linear and continuous map of X to Y , equipped with its usual norm of the uniform convergence on the unit sphere of X. If U is an open subset of X, and F : U → Y is a Fr´echet-differentiable map in U , we denote the differential of F by dF . The inverse function of an invertible function f is denoted by f (−1) , while the reciprocal of a function g is denoted by g −1 . Let Ω ⊆ Rn . Then Ω denotes the closure of Ω in Rn , ∂Ω denotes the boundary of Ω , and νΩ denotes the outward unit normal to ∂Ω . Let Ω be an open subset of Rn and m ∈ N \ {0}. The space of m times continuously differentiable real-valued function on Ω is denoted by C m (Ω ). Let r ∈ N \ {0}, f ∈ (C m (Ω ))r . The sth component of f is denoted by fs and the gradient matrix of f is denoted by ∇f . |η| f Let η = (η1 , . . . , ηn ) ∈ Nn and |η| = η1 + · · · + ηn . Then Dη f ≡ ∂xη1∂,...,∂x ηn . If r = 1, the Hessian 1

n

matrix of the second-order partial derivatives of f is denoted by D2 f . The subspace of C m (Ω ) of those functions f such that f and its derivatives Dη f of order |η| ≤ m can be extended with continuity to Ω is denoted C m (Ω ). The subspace of C m (Ω ) whose functions have m-the order derivatives that are H¨older m,α 0,α continuous with exponent { α ∈ ]0, 1[ is denoted C } (Ω ). If f ∈ C (Ω ), then its H¨older constant is |f (x)−f (y)| defined as |f : Ω |α ≡ sup : x, y ∈ Ω , x ̸= y . If Ω is open and bounded, then the space C m,α (Ω ), |x−y|α ∑ equipped with its usual norm ∥f ∥C m,α (Ω) ≡ ∥f ∥C m (Ω) + |η|=m |Dη f : Ω |α , is a Banach space. We denote m,α by Cloc (Rn \ Ω ) the space of functions on Rn \ Ω whose restriction to U belongs to C m,α (U ) for all open

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m,α bounded subsets U of Rn \ Ω . On Cloc (Rn \ Ω ) we consider the natural structure of Fr´echet space. Finally we set m,α Charm (Ω ) ≡ {u ∈ C m,α (Ω ) ∩ C 2 (Ω ) : ∆u = 0 in Ω }.

We say that a bounded open subset of Rn is of class C m,α if it is a manifold with boundary imbedded in Rn of class C m,α . In particular, if Ω is a C 1,α subset of Rn , then ∂Ω is a C 1,α sub-manifold of Rn of co-dimension 1. If M is a C m,α sub-manifold of Rn of dimension d ≥ 1, we define the space C m,α (M ) by exploiting a finite local parametrization. Namely, we take a finite open covering U1 , . . . , Uk of M and C m,α local parametrization maps γl : Bd (0, 1) → Ul with l = 1, . . . , k and we say that ϕ ∈ C m,α (M ) if and only if ϕ ◦ γl ∈ C m,α (Bd (0, 1)) for all l = 1, . . . , k. Then for all ϕ ∈ C m,α (M ) we define ∥ϕ∥C m,α (M ) ≡

k ∑

∥ϕ ◦ γl ∥C m,α (B

d (0,1))

.

l=1

One verifies that different C m,α finite atlases define the same space C m,α (M ) and equivalent norms on it. We retain the standard notion for the Lebesgue spaces Lp , p ≥ 1. If Ω is of class C 1,α , we denote by dσ the area element on ∂Ω . If Z is a subspace of L1 (∂Ω ), then we set { } ∫ Z0 ≡ f ∈ Z : f dσ = 0 . (5) ∂Ω

3. Classical results of potential theory In this section we present some classical results of potential theory. For the proofs we refer to Folland [11], Gilbarg and Trudinger [12], Schauder [23], and to the references therein. Definition 3.1. We denote by Sn the function from Rn \ {0} to R defined by 2−n

Sn (x) ≡

|x| (2 − n)sn

∀x ∈ Rn \ {0},

where sn denotes the (n − 1)-dimensional measure of ∂Bn (0, 1). Sn is well known to be a fundamental solution of the Laplace operator. Now let Ω be an open bounded connected subset of Rn of class C 1,α . Definition 3.2. We denote by wΩ [µ] the double layer potential with density µ given by ∫ wΩ [µ](x) ≡ − νΩ (y) · ∇Sn (x − y)µ(y) dσy ∀x ∈ Rn ∂Ω

for all µ ∈ C 1,α (∂Ω ). We denote by W∂Ω the boundary integral operator which takes µ ∈ C 1,α (∂Ω ) to the function W∂Ω [µ] defined by ∫ W∂Ω [µ](x) ≡ − νΩ (y) · ∇Sn (x − y)µ(y) dσy ∀x ∈ ∂Ω . ∂Ω

It is well known that, if µ ∈ C 1,α (∂Ω ), then wΩ [µ]|Ω admits a unique continuous extension to Ω , which + we denote by wΩ [µ], and wΩ [µ]|Rn \Ω admits a unique continuous extension to Rn \ Ω , which we denote by − wΩ [µ].

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

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In the following Theorem 3.3 we summarize classical results in potential theory. Theorem 3.3.

The following statements hold.

(i) Let µ ∈ C 1,α (∂Ω ). Then the function wΩ [µ] is harmonic in Rn \ ∂Ω and at infinity. Moreover, we have the following jump relations ) ( 1 ± wΩ ∀x ∈ ∂Ω ; [µ](x) = ± I + W∂Ω [µ](x) 2 + [µ] is linear and continuous and the map from (ii) The map from C 1,α (∂Ω ) to C 1,α (Ω ) which takes µ to wΩ 1,α − 1,α n C (∂Ω ) to Cloc (R \ Ω ) which takes µ to wΩ [µ] is linear and continuous; (iii) The map which takes µ to W∂Ω [µ] is continuous from C 1,α (∂Ω ) to itself; (iv) If Rn \ Ω is connected, then the operator 12 I + W∂Ω is an isomorphism from C 1,α (∂Ω ) to itself.

4. An existence result for the solutions of problem (2) In this section we recall some results of [20] on the existence of a family of solutions for problem (2). In what follows uo denotes the unique solution in C 1,α (Ω o ) of the interior Dirichlet problem in Ω o with boundary datum f o , namely { ∆uo = 0 in Ω o , uo = f o on ∂Ω o . Then we have the following Proposition 4.1, where harmonic functions in Ω (ϵ) and ϵΩ i are represented in terms of uo , double layer potentials with appropriate densities, and a suitable restriction of the fundamental solution Sn (cf. [20, Prop. 5.1]). Proposition 4.1.

Let ϵ ∈]0, ϵ0 [. The map (Uϵo [·, ·, ·, ·], Uϵi [·, ·, ·, ·])

1,α 1,α from C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to Charm (Ω (ϵ)) × Charm (ϵΩ i ) which takes (ϕo , ϕi , ζ, ψ i ) to the pair of functions (Uϵo [ϕo , ϕi , ζ, ψ i ], Uϵi [ϕo , ϕi , ζ, ψ i ])

defined by Uϵo [ϕo , ϕi , ζ, ψ i ](x) [ ( · )] − + o ≡ uo (x) + ϵwΩ (x) + ϵn−1 ζ Sn (x) ϕi o [ϕ ](x) + ϵw ϵΩ i ϵ [ ( · )] + i Uϵi [ϕo , ϕi , ζ, ψ i ](x) ≡ ϵwϵΩ (x) + ζ i i ψ ϵ

∀x ∈ Ω (ϵ),

(6)

∀x ∈ ϵΩ i ,

is bijective. We recall that C 1,α (∂Ω i )0 is the subspace of C 1,α (∂Ω i ) consisting of the functions with zero integral mean on ∂Ω i (cf. (5)). The following Lemma 4.2 provides an isomorphism between C 1,α (∂Ω i )0 × R and C 1,α (∂Ω i ) (cf. [20, Lemma 4.2]). Lemma 4.2.

The map from C 1,α (∂Ω i )0 × R to C 1,α (∂Ω i ) which takes (µ, ξ) to the function ( ) 1 J[µ, ξ] ≡ − I + W∂Ω i [µ] + ξ Sn |∂Ω i 2

is an isomorphism.

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

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Let F be as in (1). We indicate by ∂ϵ F and ∂ζ F the partial derivative of F with respect to the first and the last argument, respectively. We shall exploit the following assumptions: There exist ζ i ∈ R such that F (0, ·, ζ i ) = uo (0) and (∂ζ F )(0, ·, ζ i ) is constant and positive.

(7)

and For each t ∈ ∂Ω i fixed, the map from ] − ϵ0 , ϵ0 [ × R to R which takes (ϵ, ζ) to F (ϵ, t, ζ) is of class C 2 .

(8)

Then we have the following Lemma 4.3, concerning the Taylor expansion of the functions uo and F (cf. [20, Lemmas 5.2, 5.3]). Lemma 4.3.

Let (7) and (8) hold true. Let a, b ∈ R. Then F (ϵ, t, a + ϵb) = F (0, t, a) + ϵ(∂ϵ F )(0, t, a) + ϵb(∂ζ F )(0, t, a) + ϵ2 F˜ (ϵ, t, a, b),

for all (ϵ, t) ∈] − ϵ0 , ϵ0 [×∂Ω i , where F˜ (ϵ, t, a, b) ≡

∫

1

(1 − τ ){(∂ϵ2 F )(τ ϵ, t, a + τ ϵb)

0

+ 2b(∂ϵ ∂ζ F )(τ ϵ, t, a + τ ϵb) + b2 (∂ζ2 F )(τ ϵ, t, a + τ ϵb)} dτ. Moreover ˜o (ϵ, t) uo (ϵt) − F (0, t, ζ i ) = ϵ t · ∇uo (0) + ϵ2 u for all (ϵ, t) ∈] − ϵ0 , ϵ0 [×∂Ω i , where o

∫

1

(1 − τ )

˜ (ϵ, t) ≡ u 0

n ∑

ti tj (∂xi ∂xj uo )(τ ϵt) dτ .

i,j=1

Then we introduce a notation for the superposition operators. Definition 4.4. If H is a function from ] − ϵ0 , ϵ0 [ × ∂Ω i × R to R, then we denote by NH the (nonlinear nonautonomous) superposition operator which takes a pair (ϵ, v) consisting of a real number ϵ ∈ ] − ϵ0 , ϵ0 [ and of a function v from ∂Ω i to R to the function NH (ϵ, v) defined by NH (ϵ, v)(t) ≡ H(ϵ, t, v(t))

∀t ∈ ∂Ω i .

Here the letter “N ” stands for “Nemytskii operator”. Having introduced Definition 4.4, we can now formulate the following assumption on the functions F and G of (1): For all (ϵ, v) ∈ ] − ϵ0 , ϵ0 [ × C 1,α (∂Ω i ) we have NF (ϵ, v) ∈ C 1,α (∂Ω i ) and NG (ϵ, v) ∈ C 0,α (∂Ω i ). Moreover, the superposition operator NF is real analytic from ] − ϵ0 , ϵ0 [ × C 1,α (∂Ω i ) to C 1,α (∂Ω i ) and the superposition

(9)

operator NG is real analytic from ] − ϵ0 , ϵ0 [ × C 1,α (∂Ω i ) to C 0,α (∂Ω i ). Then, for real analytic superposition operators we have the following Proposition 4.5 (cf. Lanza de Cristoforis [4, Prop 5.3]).

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

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Proposition 4.5. If H is a function from ] − ϵ0 , ϵ0 [×∂Ω i × R to R such that the superposition operator NH is real analytic from ] − ϵ0 , ϵ0 [×C 1,α (∂Ω i ) to C 1,α (∂Ω i ), then the partial differential of NH with respect to the second variable v, computed at the point (ϵ, v) ∈] − ϵ0 , ϵ0 [×C 1,α (∂Ω i ), is the linear operator dv NH (ϵ, v) defined by dv NH (ϵ, v).˜ v = N(∂ζ H) (ϵ, v)˜ v ∀˜ v ∈ C 1,α (∂Ω i ). (10) The same result holds replacing the domain and the target space of the operator NH with ]−ϵ0 , ϵ0 [×C 0,α (∂Ω i ) and C 0,α (∂Ω i ), respectively, and using functions v, v˜ ∈ C 0,α (∂Ω i ) in (10). In what follows we will exploit an auxiliary map M = (M1 , M2 , M3 ) from ] − ϵ0 , ϵ0 [ × C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ) defined by ) ( 1 M1 [ϵ, ϕo , ϕi , ζ, ψ i ](x) ≡ I + W∂Ω o [ϕo ](x) 2 ∫ − ϵn−1 νΩ i (y) · ∇Sn (x − ϵy)ϕi (y) dσy + ϵn−2 Sn (x)ζ ∀x ∈ ∂Ω o , ∂Ω i ) ( 1 o i i o o M2 [ϵ, ϕ , ϕ , ζ, ψ ](t) ≡ t · ∇u (0) + ϵ˜ u (ϵ, t) + − I + W∂Ω i [ϕi ](t) 2 + o i + ζ Sn (t) + wΩ o [ϕ ](ϵt) − (∂ϵ F )(0, t, ζ ) − (∂ζ F )(0, t, ζ i ) ( ) ( ( ) ) 1 1 × I + W∂Ω i [ψ i ](t) − ϵF˜ ϵ, t, ζ i , I + W∂Ω i [ψ i ](t) ∀t ∈ ∂Ω i , 2 2 ( − + o i M3 [ϵ, ϕo , ϕi , ζ, ψ i ](t) ≡ νΩ i (t) · ∇uo (ϵt) + ϵ∇wΩ o [ϕ ](ϵt) + ∇w i [ϕ ](t) Ω ) ( ( ) ) 1 + i + ∇Sn (t)ζ − ∇wΩ ϵ, t, ϵ I + W∂Ω i [ψ i ](t) + ζ i i [ψ ](t) − G 2 ∀t ∈ ∂Ω i , for all (ϵ, ϕo , ϕi , ζ, ψ i ) ∈ ] − ϵ0 , ϵ0 [ × C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ). In Theorem 4.6 we summarize some results of [20] on the operator M (cf. [20, Prop. 7.1, 7.5, 7.6, Lem. 7.7, Thm. 7.8]). Theorem 4.6.

Let assumptions (7)–(9) hold. Then the following statements hold.

(i) The map M is real analytic from ] − ϵ0 , ϵ0 [×C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ). (ii) Let ϵ ∈]0, ϵ0 [ and (ϕo , ϕi , ζ, ψ i ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ). Then the pair (uoϵ [ϕo , ϕi , ζ, ψ i ], uiϵ [ϕo , ϕi , ζ, ψ i ]) defined by (6) is a solution of (2) if and only if M [ϵ, ϕo , ϕi , ζ, ψ i ] = (0, 0, 0) .

(11)

(iii) The equation M [0, ϕo , ϕi , ζ, ψ i ] = (0, 0, 0) has a unique solution (ϕo0 , ϕi0 , ζ0 , ψ0i ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ). (iv) The partial differential of M with respect to (ϕo , ϕi , ζ, ψ i ) evaluated at (0, ϕo0 , ϕi0 , ζ0 , ψ0i ), which we denote by ∂(ϕo ,ϕi ,ζ,ψi ) M [0, ϕo0 , ϕi0 , ζ0 , ψ0i ] , is an isomorphism from C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i ) to C 1,α (∂Ω o )×C 1,α (∂Ω i )×C 0,α (∂Ω i ).

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

9

(v) There exist ϵ′ ∈]0, ϵ0 [, an open neighborhood U0 of (ϕo0 , ϕi0 , ζ0 , ψ0i ) in C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ), and a real analytic map (Φ o , Φ i , Z, Ψ i ) : ] − ϵ′ , ϵ′ [→ U0 such that the set of zeros of M in ] − ϵ′ , ϵ′ [×U0 coincides with the graph of (Φ o [·], Φ i [·], Z[·], Ψ i [·]). In particular, (Φ o [0], Φ i [0], Z[0], Ψ i [0]) = (ϕo0 , ϕi0 , ζ0 , ψ0i ). Then, in view of Theorems 4.6(ii) and 4.6(v), we can introduce a family of solutions {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ for problem (2). Theorem 4.7. Let assumptions (7), (8), and (9) hold true. Let ϵ′ and (Φ o [·], Φ i [·], Z[·], Ψ i [·]) be as in Theorem 4.6(v). For all ϵ ∈]0, ϵ′ [, let uoϵ (x) ≡ Uϵo [Φ o [ϵ], Φ i [ϵ], Z[ϵ], Ψ i [ϵ]](x) ∀x ∈ Ω (ϵ) , uiϵ (x) ≡ Uϵi [Φ o [ϵ], Φ i [ϵ], Z[ϵ], Ψ i [ϵ]](x)

∀x ∈ ϵΩ i ,

1,α 1,α (Ω (ϵ))×Charm with Uϵo [·, ·, ·, ·] and Uϵi [·, ·, ·, ·] defined as in (6). Then the pair of functions (uoϵ , uiϵ ) ∈ Charm (ϵΩ i ) ′ is a solution of (2) for all ϵ ∈]0, ϵ [.

5. Local uniqueness of the solution (uoϵ , uiϵ ) In this section we prove local uniqueness results for the family of solutions {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ of Theorem 4.7. We will denote by B0,r the ball in the product space C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) of radius r > 0 and centered in the 4-tuple (ϕo0 , ϕi0 , ζ0 , ψ0i ) of Theorem 4.6(iii). Namely, we set { B0,r ≡ (ϕo , ϕi , ζ, ψ i ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) : } (12) ∥ϕo − ϕo0 ∥C 1,α (∂Ω o ) + ∥ϕi − ϕi0 ∥C 1,α (∂Ω i ) + |ζ − ζ0 | + ∥ψ i − ψ0i ∥C 1,α (∂Ω i ) < r for all r > 0. Then for ϵ′ as in Theorem 4.6(v), we denote by Λ = (Λ1 , Λ2 ) the map from ] − ϵ′ , ϵ′ [ × C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) defined by ( ) 1 Λ1 [ϵ, ϕo , ϕi , ζ](x) ≡ I + W∂Ω o [ϕo ](x) 2 ∫ n−1 −ϵ νΩ i (y) · ∇Sn (x − ϵy)ϕi (y) dσy ∂Ω i

n−2

+ϵ Sn (x)ζ ( ) 1 + o Λ2 [ϵ, ϕo , ϕi , ζ](t) ≡ − I + W∂Ω i [ϕi ](t) + wΩ o [ϕ ](ϵt) 2 + Sn (t)ζ

∀x ∈ ∂Ω o ,

(13)

∀t ∈ ∂Ω i ,

for all (ϵ, ϕo , ϕi , ζ) ∈ ] − ϵ′ , ϵ′ [ × C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R. We now prove the following. Proposition 5.1. There exist ϵ′′ ∈]0, ϵ′ [ and C ∈]0, +∞[ such that the operator Λ[ϵ, ·, ·, ·] from C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) is linear continuous and invertible for all ϵ ∈] − ϵ′′ , ϵ′′ [ fixed and such that ∥Λ[ϵ, ·, ·, ·](−1) ∥L(C 1,α (∂Ω o )×C 1,α (∂Ω i ),C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R) ≤ C uniformly for ϵ ∈] − ϵ′′ , ϵ′′ [.

10

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

Proof . By the mapping properties of the double layer potential (cf. Theroems 3.3(ii) and 3.3(iii)) and of integral operators with real analytic kernels (cf. Lanza de Cristoforis and Musolino [7]) one verifies that the map from ] − ϵ′ , ϵ′ [ to L(C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R, C 1,α (∂Ω o ) × C 1,α (∂Ω i )) which takes ϵ to Λ[ϵ, ·, ·, ·] is continuous. Since the set of invertible operators is open in the space L(C 1,α (∂Ω o )×C 1,α (∂Ω i )0 × R, C 1,α (∂Ω o ) × C 1,α (∂Ω i )), to complete the proof it suffices to show that for ϵ = 0 the map which takes (ϕo , ϕi , ζ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R to Λ[0, ϕo , ϕi , ζ] ) ) ) ( (( 1 1 + o i o I + W∂Ω o [ϕ ], − I + W∂Ω i [ϕ ] + wΩ o [ϕ ](0) + Sn |∂Ω i ζ = 2 2 1,α o 1,α i ∈ C (∂Ω ) × C (∂Ω ) is invertible. To prove it, we verify that it is a bijection and then we exploit the Open Mapping Theorem. So let (ho , hi ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i ). We claim that there exists a unique (ϕo , ϕi , ζ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R such that Λ[0, ϕo , ϕi , ζ] = (ho , hi ).

(14)

Indeed, by Theorem 3.3(iv), 21 I + W∂Ω o is an isomorphism from C 1,α (∂Ω o ) into itself and there exists a unique ϕo ∈ C 1,α (∂Ω o ) that satisfies the first equation of (14). Moreover, by Lemma 4.2, the map from ( ) C 1,α (∂Ω i )0 × R to C 1,α (∂Ω i ) that takes (ϕi , ζ) to − 12 I + W∂Ω i [ϕi ] + Sn |∂Ω i ζ, is an isomorphism. Hence, there exists a unique (ϕi , ζ) ∈ C 1,α (∂Ω i )0 × R such that ) ( 1 + o − I + W∂Ω i [ϕi ] + Sn |∂Ω i ζ = hi − wΩ o [ϕ ](0). 2 Accordingly, there exists a unique (ϕi , ζ) ∈ C 1,α (∂Ω i )0 × R that satisfies the second equation of (14). Thus Λ[0, ·, ·, ·] is an isomorphism from C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) and the proof is complete. □ 5.1. A first local uniqueness result We are now ready to state our first local uniqueness result for the solution (uoϵ , uiϵ ). Theorem 5.2 is, in a sense, a consequence of an argument based on the Implicit Function Theorem for real analytic maps (see, for example, Deimling [10, Thm. 15.3]) that has been used in [20] to prove the existence of such solution. We shall see in the following Section 5.2 that the statement of Theorem 5.2 holds under much weaker assumptions. Theorem 5.2. Let assumptions (7), (8), and (9) hold true. Let ϵ′ ∈]0, ϵ0 [ be as in Theorem 4.6(v). Let {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ be as in Theorem 4.7. Then there exist ϵ∗ ∈]0, ϵ′ [ and δ ∗ ∈]0, +∞[ such that the following property holds: If ϵ ∈]0, ϵ∗ [ and (v o , v i ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ) is a solution of problem (2) with ∥v o − uoϵ ∥C 1,α (∂Ω o ) ≤ ϵδ ∗ , o

∥v (ϵ·) −  i v (ϵ·) −

uoϵ (ϵ·)∥C 1,α (∂Ω i )  uiϵ (ϵ·)C 1,α (∂Ω i )

then (v o , v i ) = (uoϵ , uiϵ ) .

(15)

∗

(16)

∗

(17)

≤ ϵδ , ≤ ϵδ ,

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

11

Proof . Let U0 be the open neighborhood of (ϕo0 , ϕi0 , ζ0 , ψ0i ) in C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) introduced in Theorem 4.6(v). We take K > 0 such that B0,K ⊆ U0 . Since (Φ o [·], Φ i [·], Z[·], Ψ i [·]) is continuous (indeed real analytic) from ] − ϵ′ , ϵ′ [ to U0 , then there exists ϵ′∗ ∈ ]0, ϵ′ [ such that (Φ o [η], Φ i [η], Z[η], Ψ i [η]) ∈ B0,K/2 ∀η ∈ ]0, ϵ′∗ [ . (18) Let ϵ′′ be as in Proposition 5.1 and let ϵ∗ ≡ min{ϵ′∗ , ϵ′′ }. Let ϵ ∈ ]0, ϵ∗ [ be fixed and let (v o , v i ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ) be a solution of problem (2) that satisfies (15), (16), and (17) for a certain δ ∗ ∈ ]0, +∞[. We show that for δ ∗ sufficiently small (v o , v i ) = (uoϵ , uiϵ ). By Proposition 4.1, there exists a unique quadruple (ϕo , ϕi , ζ, ψ i ) ∈ C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) such that v o = Uϵo [ϕo , ϕi , ζ, ψ i ] i

v =

Uϵi [ϕo , ϕi , ζ, ψ i ]

in Ω (ϵ),

(19)

ϵΩ i .

(20)

in

By (17) and by (20), we have  i   v (ϵ·) − uiϵ (ϵ·)  ∗   δ ≥  1,α i ϵ C (∂Ω )  i o i   Uϵ [ϕ , ϕ , ζ, ψ i ](ϵ·) − uiϵ (ϵ·)   =   1,α i ϵ C (∂Ω )   [ ( )]  ϵw+ ψ i · (ϵ·) + ζ i − ϵw+ [Ψ i [ϵ] ( · )] (ϵ·) − ζ i  i  ϵΩ i  ϵ ϵ ϵΩ =    1,α i ϵ C (∂Ω )  + i  + i  = wΩ i [ψ ] − wΩ [Ψ [ϵ]] . i C 1,α (∂Ω i ) By the jump relations in Theorem 3.3(i), we obtain  ( ( ) )   1 ∗  I + W i [ψ i ] − 1 I + W i [Ψ i [ϵ]] ∂Ω ∂Ω  1,α i ≤ δ .  2 2 C (∂Ω )

(21)

(22)

By Theorem 3.3(iv), the operator 21 I + W∂Ω i is a linear isomorphism from C 1,α (∂Ω i ) to itself. Then, if we denote by D the norm of its inverse, namely we set ( )(−1)   1    D ≡  I + W∂Ω i ,   2  1,α i 1,α i L(C

(∂Ω ),C

(∂Ω ))

we obtain, by (17) and by (22), that ∥ψ i − Ψ i [ϵ]∥C 1,α (∂Ω i ) ( )(−1)   1    ≤  I + W∂Ω i   2  1,α i 1,α i L(C (∂Ω ),C (∂Ω )) (  ) ( )  1  1 i i  ×  I + W∂Ω i [ψ ] − I + W∂Ω i [Ψ [ϵ]]  1,α i 2 2 C (∂Ω ) ≤ Dδ ∗ .

(23)

12

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

By (15) and (19) we have  o o i    o  Uϵ [ϕ , ϕ , ζ, ψ i ] − uoϵ   v − uoϵ     = δ∗ ≥    1,α o  1,α o  ϵ ϵ C (∂Ω ) C (∂Ω )   [ ( ) ( )]  ϵw+o [ϕo − Φ o [ϵ]] + ϵw− ϕi · − Φ i [ϵ] · + ϵn−1 (ζ − Z[ϵ]) Sn  i  Ω  ϵ ϵ ϵΩ =    1,α o ϵ   C (∂Ω ) [ (·) ( · )]  + o  − o ϕi = wΩ − Φ i [ϵ] + ϵn−2 (ζ − Z[ϵ]) Sn  . o [ϕ − Φ [ϵ]] + w ϵΩ i ϵ ϵ C 1,α (∂Ω o ) Similarly, (16) and (19) yield  o o i    o  Uϵ [ϕ , ϕ , ζ, ψ i ](ϵ·) − uoϵ (ϵ·)   v (ϵ·) − uoϵ (ϵ·)  ∗     δ ≥  1,α i =   1,α i ϵ ϵ C (∂Ω ) C (∂Ω )  [ (·) ( · )]  + o − o i = wΩ o [ϕ − Φ [ϵ]](ϵ·) + wϵΩ i ϕ − Φ [ϵ] (ϵ·) ϵ ϵ  + ϵn−2 (ζ − Z[ϵ]) Sn (ϵ·)C 1,α (∂Ω i )   − [ i ] + o o i  = wΩ i ϕ − Φ [ϵ] + wΩ o [ϕ − Φ [ϵ]](ϵ·) + (ζ − Z[ϵ]) Sn C 1,α (∂Ω i ) . Then, by (24) and (25) and by the definition of the operator Λ in (13), we deduce that  [ o ] Λ ϵ, ϕ − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ]  1,α o ≤ 2δ ∗ C (∂Ω )×C 1,α (∂Ω i )

(24)

(25)

(26)

(see also the jump relations for the double layer potential in Theorem 3.3(i)). Now let C > 0 as in the statement of Proposition 5.1. Then, by the membership of ϵ in ]0, ϵ∗ [, we have ( o ) ϕ − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ] (27) [ ] = Λ[ϵ, ·, ·, ·](−1) Λ ϵ, ϕo − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ] and, by (26) and (27), we obtain ( ) ∥ ϕo − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ] ∥C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R ≤ ∥Λ[ϵ, ·, ·, ·](−1) ∥L(C 1,α (∂Ω o )×C 1,α (∂Ω i ),C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R)  [ ] × Λ ϵ, ϕo − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ]  1,α o 1,α i C

(∂Ω )×C

(∂Ω )0 ×R

≤ 2Cδ ∗ . The latter inequality, combined with (23), yields ( ) ∥ ϕo − Φ o [ϵ], ϕi − Ψ i [ϵ], ζ − Z[ϵ], ψ i − Ψ [ϵ] ∥C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i ) ≤ (2C + D)δ ∗ .

(28)

Hence, by (18) and (28) and by a standard computation based on the triangular inequality one sees that ∥(ϕo , ϕi , ζ, ψ i ) − (ϕo0 , ϕi0 , ζ0 , ψ0i )∥C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i ) K ≤ (2C + D)δ ∗ + . 2 o i i Accordingly, in order to have (ϕ , ϕ , ζ, ψ ) ∈ B0,K , it suffices to take δ∗ <

K 2(2C + D)

in inequalities (15), (16), and (17). Then, by the inclusion B0,K ⊆ U0 and by Theorem 4.6(v), we deduce that for such choice of δ ∗ we have ( ) (ϕo , ϕi , ζ, ψ i ) = Φ o [ϵ], Ψ i [ϵ], Z[ϵ], Ψ [ϵ] and thus (v o , v i ) = (uoϵ , uiϵ ) (cf. Theorem 4.7). □

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

13

5.2. A stronger local uniqueness result In this Subsection we will see that we can weaken the assumptions of Theorem 5.2. In particular, we will prove in Theorem 5.6 that the local uniqueness of the solution can be achieved with only one condition on the trace of the function v i (ϵ·) on ∂Ω i , instead of the three conditions used in Theorem 5.2. To prove Theorem 5.6 we shall need some preliminary technical results on composition operators. 5.2.1. Some preliminary results on composition operators We begin with the following Lemma 5.3. Lemma 5.3. Let A be a function from ] − ϵ0 , ϵ0 [×Bn−1 (0, 1) × R to R. Let MA be the map which takes a pair (ϵ, ζ) ∈] − ϵ0 , ϵ0 [×R to the function MA (ϵ, ζ) defined by MA (ϵ, ζ)(z) ≡ A(ϵ, z, ζ)

∀z ∈ Bn−1 (0, 1).

(29)

Let m ∈ {0, 1}. If MA (ϵ, ζ) ∈ C m,α (Bn−1 (0, 1)) for all (ϵ, ζ) ∈]−ϵ0 , ϵ0 [×R and if the map MA is real analytic from ]−ϵ0 , ϵ0 [×R to C m,α (Bn−1 (0, 1)), then for every open bounded interval J of R and every compact subset E of ] − ϵ0 , ϵ0 [ there exists C > 0 such that sup ∥A(ϵ, ·, ·)∥C m,α (B

n−1 (0,1)×J )

ϵ∈E

≤ C.

(30)

Proof . We first prove the statement of Lemma 5.3 for m = 0. If MA is real analytic from ] − ϵ0 , ϵ0 [ × R to ˜ ∈ ] − ϵ0 , ϵ0 [ × R there exist M ∈ ]0, +∞[, ρ ∈ ]0, 1[, and a family of ϵ, ζ) C 0,α (Bn−1 (0, 1)), then for every (˜ coefficients {ajk }j,k∈N ⊂ C 0,α (Bn−1 (0, 1)) such that ∥ajk ∥C 0,α (B and MA (ϵ, ζ)(·) =

∞ ∑

n−1 (0,1))

≤M

˜j ajk (·)(ϵ − ϵ˜)k (ζ − ζ)

( )k+j 1 ρ

∀j, k ∈ N,

∀(ϵ, ζ) ∈ ]˜ ϵ − ρ, ϵ˜ + ρ[ × ]ζ˜ − ρ, ζ˜ + ρ[ ,

(31)

(32)

j,k=0

where ρ is less than or equal to the radius of convergence of the series in (32). Now let J ⊂ R be open and bounded and E ⊂ ] − ϵ0 , ϵ0 [ be compact. Since the product J × E is compact, a standard finite covering argument shows that in order to prove (30) for m = 0 it suffices to find a uniform upper bound (independent ˜ for the quantity of ϵ˜ and ζ) sup ρ

ρ

ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ]

∥A(ϵ, ·, ·)∥C 0,α (B

˜ ρ ˜ ρ n−1 (0,1)×[ζ− 4 ,ζ+ 4 ])

.

By (29), (31), and (32) we have sup ρ

ρ

ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ]

∥A(ϵ, ·, ·)∥C 0 (B

˜ ρ ˜ ρ n−1 (0,1)×[ζ− 4 ,ζ+ 4 ])

≤

sup

∞ ∑

ρ ρ ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ] j,k=0

˜ j∥ 0 ∥ajk (·)(ϵ − ϵ˜)k (· − ζ) C (B

˜ ρ ˜ ρ n−1 (0,1)×[ζ− 4 ,ζ+ 4 ])

( )j+k ( ) ( ) ( )j+k ∞ ∞ ∑ ∑ 1 1 ρ k ρ j ≤ M = M = 4M ρ 2 2 2 j,k=0

j,k=0

(33)

14

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

for all l = 1, . . . , m. Then inequality (33) yields an estimate of the C 0 norm of A. To complete the proof of (30) for m = 0 we have now to study the H¨ older constant of A(ϵ, ·, ·) on Bn−1 (0, 1) × [ζ˜ − ρ4 , ζ˜ + ρ4 ]. To do ′ ′′ ′ ′′ ˜ ϵ − ρ2 , ϵ˜ + ρ2 ], and we consider the difference so, we take z , z ∈ Bn−1 (0, 1), ζ , ζ ∈ [ζ − ρ4 , ζ˜ + ρ4 ], and ϵ ∈ [˜ ˜ j − ajk (z ′′ )(ϵ − ϵ˜)k (ζ ′′ − ζ) ˜ j |. |ajk (z ′ )(ϵ − ϵ˜)k (ζ ′ − ζ)

(34)

˜ j inside For j ≥ 1 and k ≥ 0 we argue as follows: we add and subtract the term ajk (z ′′ )(ϵ − ϵ˜)k (ζ ′ − ζ) the absolute value in (34), we use the triangular inequality to split the difference in two terms and then we exploit the membership of ajk in C 0,α (Bn−1 (0, 1)) and an argument based on the Taylor expansion at the ˜ j . Doing so we show that (34) is first order for the function from [ζ˜ − ρ4 , ζ˜ + ρ4 ] to R that takes ζ to (ζ − ζ) less than or equal to k ˜ j + |ajk (z ′′ )||ϵ − ϵ˜|k |(ζ ′ − ζ) ˜ j − (ζ ′′ − ζ) ˜ j| |ajk (z ′ ) − ajk (z ′′ )||ϵ − ϵ˜| |ζ ′ − ζ|

≤ ∥ajk ∥C 0,α (B

α k ˜j |z ′ − z ′′ | |ϵ − ϵ˜| |ζ ′ − ζ| ( ) k ˜ j−1 |ζ ′ − ζ ′′ | |ϵ − ϵ˜| j|ζ − ζ| (0,1))

n−1 (0,1))

+ ∥ajk ∥C 0,α (B

n−1

(35)

˜ ≤ ρ , and |ζ − ζ| ˜ ≤ ρ , and for a suitable ζ ∈ [ζ˜ − ρ4 , ζ˜ + ρ4 ]. Then by (31), by inequalities |ϵ − ϵ˜| ≤ ρ2 , |ζ ′ − ζ| 4 4 by a straightforward computation we see that the right hand side of (35) is less than or equal to ( )j+k ( )j+k ( ) ( ) ( ) ( ) 1 1 ρ j−1 ′ ρ k α ρ k ρ j M +M j |ζ − ζ ′′ | |z ′ − z ′′ | ρ 2 4 ρ 2 4 (36) ( )j ( )j+k ( )j+k 1 1 1 ′ ′′ α −1 j ′ ′′ 1−α ′ ′′ α =M |z − z | + 4M ρ |ζ − ζ | |ζ − ζ | . 2 2 2j 2 1−α 1−α ≤ (ρ/2) and since Now, since ζ ′ and ζ ′′ are taken in the interval [ζ˜ − ρ4 , ζ˜ + ρ4 ] we have |ζ ′ − ζ ′′ | ′ ′′ 1−α ρ ∈ ]0, 1[ and α ∈ ]0, 1[, we deduce that |ζ − ζ | ≤ 1. Moreover, since j ≥ 1, we have j/2j ≤ 1 and (1/2)j < 1. It follows that the right hand side of (36) is less than or equal to ( )j+k ( )j+k 1 1 α α M |z ′ − z ′′ | + 4M ρ−1 |ζ ′ − ζ ′′ | 2 2 (37) ( )j+k ( ′ ) 1 ′′ α ′ ′′ α −1 ≤ 4M ρ |z − z | + |ζ − ζ | 2

(also note that ρ−1 > 1). Finally, by inequality α

α

aα + bα ≤ 21− 2 (a2 + b2 ) 2 , which holds for all a, b > 0, we deduce that the right hand side of (37) is less than or equal to ( )j+k α 1 α 23− 2 M ρ−1 |(z ′ , ζ ′ ) − (z ′′ , ζ ′′ )| , 2

(38)

where |(z ′ , ζ ′ ) − (z ′′ , ζ ′′ )| denotes the Euclidean norm of the vector (z ′ , ζ ′ ) − (z ′′ , ζ ′′ ) in Rn−1 × R = Rn . Then, by (35)–(38), we obtain that ˜ j − ajk (z ′′ )(ϵ − ϵ˜)k (ζ ′′ − ζ) ˜ j| |ajk (z ′ )(ϵ − ϵ˜)k (ζ ′ − ζ) ( )j+k α 1 α |(z ′ , ζ ′ ) − (z ′′ , ζ ′′ )| ≤ 23− 2 M ρ−1 2

(39)

for all j ≥ 1, k ≥ 0, and ϵ ∈ [˜ ϵ − ρ2 , ϵ˜ + ρ2 ]. Now, for every ϵ ∈ [˜ ϵ − ρ2 , ϵ˜ + ρ2 ] we denote by a ˜jk,ϵ the function [ ] ρ ρ a ˜jk,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + →R 4 4 (40) ˜j. (z, ζ) ↦→ a ˜jk,ϵ (z, ζ) ≡ ajk (z)(ϵ − ϵ˜)k (ζ − ζ)

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

Then inequality (39) readily implies that ( )j+k ⏐ [ ρ ˜ ρ ]⏐⏐ 1 ⏐ 3− α −1 ˜ 2 ajk,ϵ : Bn−1 (0, 1) × ζ − , ζ + ⏐ ≤ 2 Mρ ⏐˜ 4 4 α 2 [ ρ ρ] , ∀j ≥ 1 , k ≥ 0 , ϵ ∈ ϵ˜ − , ϵ˜ + 2 2 which in turn implies that ∞ ⏐ [ ∑ ρ ρ ]⏐⏐ ⏐ sup ajk,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ⏐˜ ρ ρ 4 4 α ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ] j=1,k=0 ( )j+k ∞ ∑ α α 1 ≤ 23− 2 M ρ−1 ≤ 24− 2 M ρ−1 . 2

15

(41)

j=1,k=0

We now turn to consider (34) in the case where j = 0 and k ≥ 0. In such case, one verifies that the quantity in (34) is less than or equal to ∥a0k ∥C 0,α (B

α

k

n−1 (0,1))

|ϵ − ϵ˜| |z ′ − z ′′ | ,

which, by (31) and by inequality |ϵ − ϵ˜| ≤ ρ2 , is less than or equal to ( )k ( ) ( )k 1 ρ k ′ 1 α α M |z − z ′′ | = M |z ′ − z ′′ | . ρ 2 2 Hence, for a ˜0k,ϵ defined as in (40) (with j = 0) we have ( )k ⏐ [ ρ ρ ]⏐⏐ 1 ⏐ a0k,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ≤ M ⏐˜ 4 4 α 2 [ ρ] ρ ∀k ≥ 0 , ϵ ∈ ϵ˜ − , ϵ˜ + , 2 2 which implies that ∞ ⏐ [ ∑ ρ ρ ]⏐⏐ ⏐ a0k,ϵ : Bn−1 (0, 1) × ζ˜ + , ζ˜ + ⏐ sup ⏐˜ ρ ρ 4 4 α ϵ+ 2 ] k=0 ϵ∈[˜ ϵ− 2 ,˜ ( )k ∞ ∑ 1 ≤ M = 2M . 2 k=0

Finally, by (32), (33), (41), and (42) we obtain sup ρ

ρ

ϵ+ 2 ] ϵ∈[˜ ϵ− 2 ,˜

=

∥A(ϵ, ·, ·)∥C 0,α (B sup ρ

ρ

ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ]

˜ ρ ˜ ρ n−1 (0,1)×[ζ+ 4 ,ζ+ 4 ])

∥A(ϵ, ·, ·)∥C 0 (B

˜ ρ ˜ ρ n−1 (0,1)×[ζ+ 4 ,ζ+ 4 ])

⏐ [ ρ ρ ]⏐⏐ ⏐ ⏐A(ϵ, ·, ·) : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ρ ρ 4 4 α ϵ+ 2 ] ϵ∈[˜ ϵ− 2 ,˜ ∞ [ ∑ ⏐⏐ ρ ρ ]⏐⏐ ≤ 4M + sup ajk,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ⏐˜ ρ ρ 4 4 α ϵ∈[˜ ϵ− ,˜ ϵ+ ] +

sup

2

= 4M +

2 j,k=0

sup ρ

ρ

∞ ⏐ [ ∑ ρ ρ ]⏐⏐ ⏐ ajk,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ⏐˜ 4 4 α

ϵ∈[˜ ϵ− 2 ,˜ ϵ+ 2 ] j=1,k=0

+

∞ ⏐ [ ∑ ρ ]⏐⏐ ρ ⏐ a0k,ϵ : Bn−1 (0, 1) × ζ˜ − , ζ˜ + ⏐ ⏐˜ ρ ρ 4 4 α ϵ∈[˜ ϵ− ,˜ ϵ+ ]

sup 2

≤ 4M + 2

2 k=0

4− α 2

M ρ−1 + 2M . α

We deduce that (30) for m = 0 holds with C = 6M + 24− 2 M ρ−1 .

(42)

16

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

We now assume that MA is real analytic from ] − ϵ0 , ϵ0 [ × R to the space C 1,α (Bn−1 (0, 1)) and we prove (30) for m = 1. To do so we will exploit the (just proved) statement of Lemma 5.3 for m = 0. We begin by observing that, since the imbedding of C 1,α (Bn−1 (0, 1)) into C 0,α (Bn−1 (0, 1)) is linear and continuous, the map MA is real analytic from ] − ϵ0 , ϵ0 [ × R to C 0,α (Bn−1 (0, 1)). Hence, by Lemma 5.3 for m = 0 and by the continuity of the imbedding of C 0,α (Bn−1 (0, 1) × J ) into C 0 (Bn−1 (0, 1) × J ) we deduce that sup ∥A(ϵ, ·, ·)∥C 0 (B ϵ∈E

n−1 (0,1)×J )

≤ C1 .

(43)

Moreover, since differentials of real analytic maps are real analytic, we have that the map M∂ζ A = ∂ζ MA which takes (ϵ, ζ) to ∂ζ A(ϵ, ·, ζ) is real analytic from ] − ϵ0 , ϵ0 [ × R to C 1,α (Bn−1 (0, 1)), and thus from ] − ϵ0 , ϵ0 [ × R to C 0,α (Bn−1 (0, 1)). By Lemma 5.3 for m = 0 it follows that sup ∥∂ζ A(ϵ, ·, ·)∥C 0,α (B ϵ∈E

n−1 (0,1)×J )

≤ C2 ,

(44)

for some C2 > 0. Finally, we observe that the map ∂z from C 1,α (Bn−1 (0, 1)) to C 0,α (Bn−1 (0, 1)) that takes a function f to ∂z f is linear and continuous. Then, the map M∂z A which takes (ϵ, ζ) to the function ∂z A(ϵ, z, ζ)

∀z ∈ Bn−1 (0, 1)

is the composition of MA and ∂z . Namely, we can write M∂z A = ∂z ◦ MA . Since MA is real analytic from ] − ϵ0 , ϵ0 [ × R to C 1,α (Bn−1 (0, 1)), it follows that M∂z A is real analytic from ] − ϵ0 , ϵ0 [ × R to C 0,α (Bn−1 (0, 1)). Hence Lemma 5.3 for m = 0 implies that there exists C3 > 0 such that sup ∥∂z A(ϵ, ·, ·)∥C 0,α (B ϵ∈E

n−1 (0,1)×J )

≤ C3 .

(45)

Now, the validity of (30) for m = 1 is a consequence of (43), (44), and (45). □ In the sequel we will exploit Schauder spaces over suitable subsets of ∂Ω i × R. We observe indeed that for all open bounded intervals J of R, the product ∂Ω i × J is a compact sub-manifold (with boundary) of co-dimension 1 in Rn ×R = Rn+1 and accordingly, we can define the spaces C 0,α (∂Ω i ×J ) and C 1,α (∂Ω i ×J ) by exploiting a finite atlas (see Section 2). ˜B be the map which takes a pair Lemma 5.4. Let B be a function from ] − ϵ0 , ϵ0 [×∂Ω i × R to R. Let N ˜B (ϵ, ζ) defined by (ϵ, ζ) ∈] − ϵ0 , ϵ0 [×R to the function N ˜B (ϵ, ζ)(t) ≡ B(ϵ, t, ζ) N

∀t ∈ ∂Ω i .

˜B (ϵ, ζ) ∈ C m,α (∂Ω i ) for all (ϵ, ζ) ∈] − ϵ0 , ϵ0 [×R and the map N ˜B is real analytic from Let m ∈ {0, 1}. If N m,α i ] − ϵ0 , ϵ0 [×R to C (∂Ω ), then for every open bounded interval J of R and every compact subset E of ] − ϵ0 , ϵ0 [ there exists C > 0 such that sup ∥B(ϵ, ·, ·)∥C m,α (∂Ω i ×J ) ≤ C .

(46)

ϵ∈E

Proof . Since ∂Ω i is a compact sub-manifold of class C 1,α in Rn , there exist a finite open covering U1 , . . . , Uk of ∂Ω i and C 1,α local parametrization maps γl : Bn−1 (0, 1) → Ul with l = 1, . . . , k. Moreover, we can assume (−1) without loss of generality that the norm of C m,α (∂Ω i ) is defined on the atlas {(Ul , γl )}l=1,...,k and the

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

17

(−1)

, idJ ))}l=1,...,k , where idJ is the identity map norm of C m,α (∂Ω i × J ) is defined on the atlas {(Ul × J , (γl from J to itself. Then, in order to prove (46) it suffices to show that sup ∥B(ϵ, γl (·), ·)∥C m,α (B ϵ∈E

n−1 (0,1)×J )

≤C

∀l ∈ {1, . . . , k}

(47)

for some C > 0. Let l ∈ {1, . . . , k} and let A be the map from ] − ϵ0 , ϵ0 [ × Bn−1 (0, 1) × R to R defined by A(ϵ, z, ζ) = B(ϵ, γl (z), ζ)

∀(ϵ, z, ζ) ∈ ] − ϵ0 , ϵ0 [ × Bn−1 (0, 1) × R .

(48)

Then, with the notation of Lemma 5.3, we have ( ) ˜B (ϵ, ζ) MA (ϵ, ζ) = γl∗ N , |U l

(

˜B (ϵ, ζ) where γl∗ N |U

) l

˜B (ϵ, ζ) by the parametrization γl . Since the is the pull back of the restriction N |U l

restriction map from C m,α (∂Ω i ) to C m,α (Ul ) and the pullback map γl∗ from C m,α (Ul ) to C m,α (Bn−1 (0, 1)) are linear and continuous and since NB is real analytic from ] − ϵ0 , ϵ0 [ × R to C m,α (∂Ω i ), it follows that the map MA is real analytic from ] − ϵ0 , ϵ0 [ × R to C m,α (Bn−1 (0, 1)). Then Lemma 5.3 implies that sup ∥A(ϵ, ·, ·)∥C m,α (B

n−1 (0,1)×J )

ϵ∈E

≤C

(49)

for some C > 0. Now the validity of (47) follows from (48) and (49). The proof is complete.

□

5.2.2. The auxiliary maps N and S In the proof of our main Theorem 5.6 we will exploit two auxiliary maps, which we denote by N and S and are defined as follows. Let ϵ′ be as in Theorem 4.6(v). We denote by N = (N1 , N2 , N3 ) the map from ] − ϵ′ , ϵ′ [ × C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ) defined by ( ) 1 o i i I + W∂Ω o [ϕo ](x) N1 [ϵ, ϕ , ϕ , ζ, ψ ](x) ≡ 2 ∫ − ϵn−1 νΩ i (y) · ∇Sn (x − ϵy)ϕi (y) dσy + ϵn−2 ζSn (x) ∀x ∈ ∂Ω o , (50) i ∂Ω ) ( 1 N2 [ϵ, ϕo , ϕi , ζ, ψ i ](t) ≡ − I + W∂Ω i [ϕi ](t) + ζ Sn (t) 2 ( ) 1 + o i + wΩ o [ϕ ](ϵt) − (∂ζ F )(0, t, ζ ) I + W∂Ω i [ψ i ](t) ∀t ∈ ∂Ω i , (51) 2 ( + o N3 [ϵ, ϕo , ϕi , ζ, ψ i ](t) ≡ νΩ i (t) · ϵ∇wΩ o [ϕ ](ϵt) ) − + i i + ∇wΩ ∀t ∈ ∂Ω i , (52) i [ϕ ](t) + ζ∇Sn (t) − ∇wΩ i [ψ ](t) for all (ϵ, ϕo , ϕi , ζ, ψ i ) ∈ ]−ϵ′ , ϵ′ [×C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i ) and we denote by S = (S1 , S2 , S3 ) the map from ] − ϵ′ , ϵ′ [ × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ) defined by S1 [ϵ, ψ i ](x) ≡ 0 i

o

o

∀x ∈ ∂Ω o ,

(53)

∀t ∈ ∂Ω i ,

(54)

∀t ∈ ∂Ω i ,

(55)

i

S2 [ϵ, ψ ](t) ≡ −t · ∇u (0) − ϵu (ϵt) + (∂ϵ F )(0, t, ζ ) ) ) ( ( 1 i i ˜ I + W∂Ω i [ψ ](t) + ϵF ϵ, t, ζ , 2 S3 [ϵ, ψ i ](t) ( ( ) ) 1 ≡ −νΩ i (t) · ∇uo (ϵt) + G ϵ, t, ϵ I + W∂Ω i [ψ i ](t) + ζ i 2

for all (ϵ, ψ i ) ∈ ] − ϵ′ , ϵ′ [ × C 1,α (∂Ω i ). For the maps N and S we have the following result.

18

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

Proposition 5.5. Let assumptions (7), (8), and (9) hold true. Then there exists ϵ′′ ∈ ]0, ϵ′ [ such that the following statements hold: (i) For all fixed ϵ ∈] − ϵ′′ , ϵ′′ [ the operator N [ϵ, ·, ·, ·, ·] is a linear homeomorphism from C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ); (ii) The map from ] − ϵ′′ , ϵ′′ [ to L(C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i ), C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i )) which takes ϵ to N [ϵ, ·, ·, ·, ·](−1) is real analytic; (iii) Equation (11) is equivalent to (ϕo , ϕi , ζ, ψ i ) = N [ϵ, ·, ·, ·, ·](−1) [S[ϵ, ψ i ]]

(56)

for all (ϵ, ϕo , ϕi , ζ, ψ i ) ∈] − ϵ′′ , ϵ′′ [×C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ). Proof . By the definition of N (cf. (50)–(52)), by the mapping properties of the double layer potential (cf. Theorem 3.3 (ii)–(iii)) and of integral operators with real analytic kernels and no singularity (see, e.g., Lanza de Cristoforis and Musolino [7]), by assumption (9) (which implies that (∂ζ F )(0, ·, ζ i ) belongs to C 1,α (∂Ω i )), and by standard calculus in Banach spaces, one verifies that the map from ] − ϵ′ , ϵ′ [ to L(C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) , C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i )) which takes ϵ to N [ϵ, ·, ·, ·, ·] is real analytic. Then one observes that N [0, ϕo , ϕi , ζ, ψ i ] = ∂(ϕo ,ϕi ,ζ,ψi ) M [0, ϕo0 , ϕi0 , ζ0 , ψ0i ].(ϕo , ϕi , ζ, ψ i ) and thus Theorem 4.6(iv) implies that N [0, ·, ·, ·, ·] is an isomorphism from C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) to C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 1,α (∂Ω i ). Since the set of invertible operators is open in L(C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ), C 1,α (∂Ω o ) × C 1,α (∂Ω i ) × C 0,α (∂Ω i )) and since the map which takes a linear invertible operator to its inverse is real analytic (cf. Hille and Phillips [13]), we deduce the validity of (i) and (ii). To prove (iii) we observe that, by the definition of N and S in (50)–(55) it readily follows that (11) is equivalent to N [ϵ, ϕo , ϕi , ζ, ψ i ] = S[ϵ, ψ i ] . Then the validity of (iii) is a consequence of (i).

□

5.2.3. The main theorem We are now ready to prove our main Theorem 5.6 on the local uniqueness of the solution (uoϵ , uiϵ ) provided by Theorem 4.7. Theorem 5.6. Let assumptions (7), (8), and (9) hold true. Let ϵ′ ∈]0, ϵ0 [ be as in Theorem 4.6(v). Let {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ be as in Theorem 4.7. Then there exist ϵ∗ ∈]0, ϵ′ [ and δ ∗ ∈]0, +∞[ such that the following property holds: If ϵ ∈]0, ϵ∗ [ and (v o , v i ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ) is a solution of problem (2) with  i  v (ϵ·) − uiϵ (ϵ·) 1,α i < ϵδ ∗ , C (∂Ω ) then (v o , v i ) = (uoϵ , uiϵ ) . Proof . • Step 1: Fixing ϵ∗ . Let ϵ′′ ∈ ]0, ϵ′ [ be as in Proposition 5.5 and let ϵ′′′ ∈ ]0, ϵ′′ [ be fixed. By the compactness of [−ϵ′′′ , ϵ′′′ ] and by the continuity of the norm in L(C 1,α (∂Ω i ) × C 0,α (∂Ω i ) × C 1,α (∂Ω o ), C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

19

× C 1,α (∂Ω i )), there exists a real number C1 > 0 such that ∥N [ϵ, ·, ·, ·, ·](−1) ∥L(C 1,α (∂Ω i )×C 0,α (∂Ω i )×C 1,α (∂Ω o ),C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i )) ≤ C1

(57)

for all ϵ ∈ [−ϵ′′′ , ϵ′′′ ] (see also Proposition 5.5(ii)). Let U0 be the open neighborhood of (ϕo0 , ϕi0 , ζ0 , ψ0i ) in C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) introduced in Theorem 4.6(v). Then we take K > 0 such that B0,K ⊆ U0 (see (12) for the definition of B0,K ). Since (Φ o [·], Φ i [·], Z[·], Ψ i [·]) is continuous (indeed real analytic) from ] − ϵ′ , ϵ′ [ to U0 , there exists ϵ∗ ∈ ]0, ϵ′′′ [ such that (Φ o [η], Φ i [η], Z[η], Ψ i [η]) ∈ B0,K/2 ⊂ U0

∀η ∈ ]0, ϵ∗ [ .

(58)

Moreover, we assume that ϵ∗ < 1. We will prove that the theorem holds for such choice of ϵ∗ . We observe that the condition ϵ∗ < 1 is not really needed in the proof but simplifies many computations. • Step 2: Planning our strategy. We suppose that there exists a pair of functions (v o , v i ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ) that is a solution of problem (2) for a certain ϵ ∈ ]0, ϵ∗ [ (fixed) and such that  i   v (ϵ·) − uiϵ (ϵ·)  ∗   (59)   1,α i ≤ δ , ϵ C (∂Ω ) for some δ ∗ ∈ ]0, +∞[. Then, by Proposition 4.1, there exists a unique quadruple (ϕo , ϕi , ζ, ψ i ) ∈ C 1,α (∂Ω o )× C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) such that v o = Uϵo [ϕo , ϕi , ζ, ψ i ] i

v =

Uϵi [ϕo , ϕi , ζ, ψ i ]

in Ω (ϵ), in ϵΩ i .

(60)

We shall show that for δ ∗ small enough we have (ϕo , ϕi , ζ, ψ i ) = (Φ o [ϵ], Φ i [ϵ], Z[ϵ], Ψ i [ϵ]) .

(61)

Indeed, if we have (61), then Theorem 4.7 would imply that (v o , v i ) = (uoϵ , vϵi ), and our proof would be completed. Moreover, to prove (61) it suffices to show that (ϕo , ϕi , ζ, ψ i ) ∈ B0,K ⊂ U0 .

(62)

In fact, in that case, both (ϵ, ϕo , ϕi , ζ, ψ i ) and (ϵ, Φ o [ϵ], Φ i [ϵ], Z[ϵ], Ψ i [ϵ]) would stay in the zero set of M (cf. Theorem 4.6(ii)) and thus (62) together with (58) and Theorem 4.6(v) would imply (61). So, our aim is now to prove that (62) holds true for a suitable choice of δ ∗ > 0. It will be also convenient to restrict our search to 0 < δ ∗ < 1. As for the condition ϵ∗ < 1, this condition on δ ∗ is not really needed, but simplifies our computations. Then to find δ ∗ and prove (62) we will proceed as follows. First we obtain an estimate for ψ i and Ψ i [ϵ] with a bound that does not depend on ϵ and δ ∗ . Then we use such estimate to show that ∥S[ϵ, ψ i ] − S[ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω o )×C 1,α (∂Ω i )×C 0,α (∂Ω i )

20

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

is smaller than a constant times δ ∗ , with a constant that does not depend on ϵ and δ ∗ . We will split the analysis for S1 , S2 , and S3 and we find convenient to study S3 before S2 . Indeed, the computations for S2 and S3 are very similar but those for S3 are much shorter and can serve better to illustrate the techniques employed. We also observe that the analysis for S2 requires the study of other auxiliary functions T1 , T2 , and T3 that we will introduce. Finally, we will exploit the estimate for ∥S[ϵ, ψ i ] − S[ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω o )×C 1,α (∂Ω i )×C 0,α (∂Ω i ) to determine δ ∗ and prove (62). • Step 3: Estimate for ψ and Ψ [ϵ]. By condition (59), by the second equality in (60), by Theorem 4.7, and by arguing as in (21) and (23) in Theorem 5.2, we obtain  ( ) ) (  1  ∗  I + W i [ψ i ] − 1 I + W i [Ψ i [ϵ]] (63) ∂Ω ∂Ω  2  1,α i ≤ δ 2 C (∂Ω ) and ( )(−1)    1   ∥ψ − Ψ [ϵ]∥C 1,α (∂Ω i ) ≤  I + W∂Ω i   1,α i 1,α i  2 L(C (  (∂Ω ),C (∂Ω )) ) ( )  1  1 ∗ i i  ×  2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]] 1,α i ≤ C2 δ , C (∂Ω ) i

where

i

( )(−1)   1    C2 ≡  I + W∂Ω i   2 

(64)

.

L(C 1,α (∂Ω i ),C 1,α (∂Ω i ))

By (58) we have ∥ψ0i − Ψ i [η]∥C 1,α (∂Ω i ) ≤

K 2

∀η ∈ ]0, ϵ∗ [.

(65)

Then, by (64) and (65), and by the triangular inequality, we see that ∥ψ i ∥C 1,α (∂Ω i ) ≤ ∥ψ0i ∥C 1,α (∂Ω i ) + ∥ψ i − Ψ i [ϵ]∥C 1,α (∂Ω i ) + ∥Ψ i [ϵ] − ψ0i ∥C 1,α (∂Ω i ) K ≤ ∥ψ0i ∥C 1,α (∂Ω i ) + C2 δ ∗ + , 2 K i i ∥Ψ [ϵ]∥C 1,α (∂Ω i ) ≤ ∥ψ0 ∥C 1,α (∂Ω i ) + . 2 Then, by taking R1 ≡ ∥ψ0i ∥C 1,α (∂Ω i ) + C2 + one verifies that

K 2

∥ψ i ∥C 1,α (∂Ω i ) ≤ R1

and R2 ≡ ∥ψ0i ∥C 1,α (∂Ω i ) + and

K 2

(and recalling that δ ∗ ∈ ]0, 1[),

∥Ψ i [ϵ]∥C 1,α (∂Ω i ) ≤ R2 .

(66)

We note here that both R1 and R2 do not depend on ϵ and δ ∗ as long they belong to ]0, ϵ∗ [ and ]0, 1[, respectively. • Step 4: Estimate for S1 . We now pass to estimate the norm ∥S[ϵ, ψ i ] − S[ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω o )×C 1,α (∂Ω i )×C 0,α (∂Ω i ) . To do so we consider separately S1 , S2 , and S3 . Since S1 = 0 (cf. definition (53)), we readily obtain that ∥S1 [ϵ, ψ i ] − S1 [ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω o ) = 0.

(67)

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

21

• Step 5: Estimate for S3 . We consider S3 before S2 because its treatment is simpler and more illustrative of the techniques used. By (55) and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi [1, Thm. 1.8]), we compute that ∥S3 [ϵ, ψ i ] − S3 [ϵ, Ψ i [ϵ]]∥C 0,α (∂Ω i )  ( ( ) )  1 i i  = G ϵ, ·, ϵ I + W∂Ω i [ψ ] + ζ 2 ) ) ( (  1 I + W∂Ω i [Ψ i [ϵ]] + ζ i  − G ϵ, ·, ϵ  0,α i 2 C (∂Ω )  ( ( ) )  1 i i = NG ϵ, ϵ 2 I + W∂Ω i [ψ ] + ζ ( ( ) )  1 − NG ϵ, ϵ I + W∂Ω i [Ψ i [ϵ]] + ζ i   0,α i 2 C (∂Ω )   ≤ dv NG (ϵ, ψ˜i )L(C 1,α (∂Ω i ),C 0,α (∂Ω i )) (  ) ( )  1  1 i i  × ϵ  I + W∂Ω i [ψ ] − I + W∂Ω i [Ψ [ϵ]]  0,α i , 2 2 C (∂Ω ) where

(68)

( ( ) ) ( ( ) ) 1 1 ψ˜i = θ ϵ I + W∂Ω i [ψ i ] + ζ i + (1 − θ) ϵ I + W∂Ω i [Ψ i [ϵ]] + ζ i , 2 2

for some θ ∈ ]0, 1[. Then, by the membership of ϵ and θ in ]0, 1[ we have (  )  1  i i i ˜  ∥ψ ∥C 1,α (∂Ω i ) ≤  I + W∂Ω i [ψ ] + ζ  2 C 1,α (∂Ω i ) (  )  1  i i  +  I + W∂Ω i [Ψ [ϵ]] + ζ  2 C 1,α (∂Ω i ) and, by setting    1  C3 ≡  I + W∂Ω i   1,α i 1,α i , 2 L(C (∂Ω ),C (∂Ω )) we obtain ∥ψ˜i ∥C 1,α (∂Ω i ) ≤ C3 ∥ψ i ∥C 1,α (∂Ω i ) + C3 ∥Ψ i [ϵ]∥C 1,α (∂Ω i ) + 2|ζ i | ≤ R,

(69)

R ≡ C3 (R1 + R2 ) + 2|ζ i |

(70)

with

which does not depend on ϵ. We wish now to estimate the operator norm   dv NG (η, ψ˜i ) 1,α i 0,α i L(C (∂Ω ),C (∂Ω )) uniformly for η ∈ ]0, ϵ∗ [. However, we cannot exploit a compactness argument on [0, ϵ∗ ] × BC 1,α (∂Ω i ) (0, R), because BC 1,α (∂Ω i ) (0, R) is not compact in the infinite dimensional space C 1,α (∂Ω i ). Then we argue as follows. We observe that, by assumption (9), the partial derivative ∂ζ G(η, t, ζ) exists for all (η, t, ζ) ∈ ] − ϵ0 , ϵ0 [ × ∂Ω i × R and, by Proposition 4.5 and Lemma 6.1(i) in the Appendix, we obtain that ∥dv NG (η, ψ˜i )∥L(C 1,α (∂Ω i ),C 0,α (∂Ω i )) ≤ ∥N∂ζ G (η, ψ˜i )∥C 0,α (∂Ω i ) ≤ ∥∂ζ G(η, ·, ψ˜i (·))∥ 0,α C

(∂Ω i )

(71)

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M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

for all η ∈ ]0, ϵ∗ [. By Proposition 6.3(ii) in the Appendix, there exists C4 > 0 such that ∥∂ζ G(η, ·, ψ˜i (·))∥C 0,α (∂Ω i ) ( ) ≤ C4 ∥∂ζ G(η, ·, ·)∥C 0,α (∂Ω×[−R,R]) 1 + ∥ψ˜i ∥α 1,α i C (∂Ω )

∀η ∈ ]0, ϵ∗ [.

(72)

˜G defined as in Lemma 5.4 (with B = G) is real Moreover, by assumption (9) one deduces that the map N 0,α i ˜G = N ˜∂ G . Hence, by analytic from ] − ϵ0 , ϵ0 [ × R to C (∂Ω ) and, by Proposition 4.5, one has that ∂ζ N ζ ∗ Lemma 5.4 (with m = 0), there exists C5 > 0 (which does not depend on ϵ ∈ ]0, ϵ [ and δ ∗ ∈ ]0, 1[) such that sup ∥∂ζ G(η, ·, ·)∥C 0,α (∂Ω i ×[−R,R]) ≤ C5 . (73) η∈[−ϵ∗ ,ϵ∗ ]

Hence, by (69), (71), (72) and (73), we deduce that ∥dv NG (ϵ, ψ˜i )∥L(C 1,α (∂Ω i ),C 0,α (∂Ω i )) ≤ C4 C5 (1 + Rα ).

(74)

By (63), (68), and (74), and by the membership of ϵ in ]0, ϵ∗ [ ⊂ ]0, 1[, we obtain that ∥S3 [ϵ, ψ i ] − S3 [ϵ, Ψ i [ϵ]]∥C 0,α (∂Ω i ) ≤ C4 C5 (1 + Rα ) δ ∗ .

(75)

• Step 6: Estimate for S2 . Finally, we consider S2 . By (54) and by the fact that ϵ ∈ ]0, 1[, we have ∥S2 [ϵ, ψ i ] − S2 [ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω i )  ( ) ) ( ( ) ) (   1 1 i i i i ˜ ˜  I + W∂Ω i [ψ ] − F ϵ, ·, ζ , I + W∂Ω i [Ψ [ϵ]]  ≤ ϵ F ϵ, ·, ζ ,  1,α i 2 2 C (∂Ω ) ∫ 1  { ≤ (1 − τ ) T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) + 2 T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) 

(76)

0

}  + T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) dτ C 1,α (∂Ω i ) , where T1 [ϵ, ψ i , Ψ i [ϵ]], T2 [ϵ, ψ i , Ψ i [ϵ]], and T3 [ϵ, ψ i , Ψ i [ϵ]] are the functions from ]0, 1[ × ∂Ω i to R defined by T1 [ϵ, ψ i , Ψ i [ϵ]](τ, t) ( ( ) ) 1 ≡ (∂ϵ2 F ) τ ϵ, t, τ ϵ I + W∂Ω i [ψ i ](t) + ζ i 2 ( ( ) ) 1 2 i i − (∂ϵ F ) τ ϵ, t, τ ϵ I + W∂Ω i [Ψ [ϵ]](t) + ζ , 2 T2 [ϵ, ψ i , Ψ i [ϵ]](τ, t) ( ) ( ( ) ) 1 1 ≡ I + W∂Ω i [ψ i ](t) (∂ϵ ∂ζ F ) τ ϵ, t, τ ϵ I + W∂Ω i [ψ i ](t) + ζ i 2 2 ( ) ( ( ) ) 1 1 − I + W∂Ω i [Ψ i [ϵ]](t)(∂ϵ ∂ζ F ) τ ϵ, t, τ ϵ I + W∂Ω i [Ψ i [ϵ]](t) + ζ i , 2 2 T3 [ϵ, ψ i , Ψ i [ϵ]](τ, t) ( ) ( ( ) ) 1 1 i 2 2 i i ≡ I + W∂Ω i [ψ ] (t) (∂ζ F ) τ ϵ, t, τ ϵ I + W∂Ω i [ψ ](t) + ζ 2 2 ( ) ( ( ) ) 1 1 − I + W∂Ω i [Ψ i [ϵ]]2 (t)(∂ζ2 F ) τ ϵ, t, τ ϵ I + W∂Ω i [Ψ i [ϵ]](t) + ζ i 2 2 for every (τ, t) ∈ ]0, 1[ × ∂Ω i .

(77)

(78)

(79)

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

23

We now want to bound the C 1,α norm with respect to the variable t ∈ ∂Ω i of (77), (78), and (79) uniformly with respect to τ ∈ ]0, 1[. • Step 6.1: Estimate for T1 . First we consider T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·). By the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi [1, Thm. 1.8]), we can estimate the C 1,α (∂Ω i ) norm of T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) (cf. (77)) as follows: ∥T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i )  ( ( ) )  1 i i  = N∂ϵ2 F τ ϵ, τ ϵ I + W∂Ω i [ψ ] + ζ 2 ) ) ( (  1 i i  I + W∂Ω i [Ψ [ϵ]] + ζ  − N∂ϵ2 F τ ϵ, τ ϵ 2 C 1,α (∂Ω i )     ≤ dv N∂ϵ2 F (τ ϵ, ψ˜1i ) 1,α i 1,α i L(C

(∂Ω ),C

(80)

(∂Ω ))

 ( ) ( )   1 1 i i  I + W [ψ ] − I + W [Ψ [ϵ]] × τϵ  i i ∂Ω ∂Ω  1,α i  2 2 C (∂Ω ) where ) ) ( ( ) ) ( ( 1 1 I + W∂Ω i [ψ i ] + ζ i + (1 − θ1 ) τ ϵ I + W∂Ω i [Ψ i [ϵ]] + ζ i , ψ˜1i = θ1 τ ϵ 2 2 for some θ1 ∈ ]0, 1[. • Step 6.2: Estimate for T2 . We now consider T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·). Adding and subtracting (

1 I + W∂Ω i 2

)

( ( ) ) 1 [Ψ i [ϵ]] (∂ϵ ∂ζ F ) τ ϵ, t, τ ϵ I + W∂Ω i [ψ i ](t) + ζ i 2

on the right hand side of (78) and using the triangular inequality, we obtain ∥T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i ) ( ) ( ( ) )  1 1 i i i ≤ I + W [ψ ] (∂ ∂ F ) τ ϵ, ·, τ ϵ I + W [ψ ] + ζ i i ϵ ζ ∂Ω ∂Ω  2 2 ( ) ( ( ) )  1 1 i i i  − I + W∂Ω i [Ψ [ϵ]] (∂ϵ ∂ζ F ) τ ϵ, ·, τ ϵ I + W∂Ω i [ψ ] + ζ  2 2 C 1,α (∂Ω i ) ( ) ( ( ) )  1 1 i i i +  2 I + W∂Ω i [Ψ [ϵ]] (∂ϵ ∂ζ F ) τ ϵ, ·, τ ϵ 2 I + W∂Ω i [ψ ] + ζ ( ) ( ( ) )  1 1 i i i  − I + W∂Ω i [Ψ [ϵ]](∂ϵ ∂ζ F ) τ ϵ, ·, τ ϵ I + W∂Ω i [Ψ [ϵ]] + ζ  . 2 2 C 1,α (∂Ω i )

(81)

By Lemma 6.1 in the Appendix and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi [1, Thm. 1.8]), we can estimate the C 1,α (∂Ω i ) norm of T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) (cf. (78) and (81)) as

24

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

follows: ∥T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i )  ( ( ) )   1 i i  (∂ ∂ F ) τ ϵ, ·, τ ϵ I + W [ψ ] + ζ ≤ 2 ∂Ω i  1,α i  ϵ ζ 2 ( C (∂Ω ) ) ( )  1  1 i i  ×  2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]] 1,α i C (∂Ω ) (  )  1  i  + 2  2 I + W∂Ω i [Ψ [ϵ]] 1,α i C (∂Ω )  ( ( ) )  1 i i N τ ϵ, τ ϵ × I + W [ψ ] + ζ i ∂Ω  ∂ϵ ∂ζ F 2 ( ( ) )  1 i i  − N∂ϵ ∂ζ F τ ϵ, τ ϵ I + W∂Ω i [Ψ [ϵ]] + ζ  2 C 1,α (∂Ω i )  ( ( ) )   1 i i  ≤ 2 (∂ϵ ∂ζ F ) τ ϵ, ·, τ ϵ 2 I + W∂Ω i [ψ ] + ζ  1,α i ( C (∂Ω ) ) ( )  1  1 i i  ×  2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]] 1,α i C (∂Ω )     i i + 2 C3 ∥Ψ [ϵ]∥C 1,α (∂Ω i ) dv N∂ϵ ∂ζ F (τ ϵ, ψ˜2 ) 1,α i 1,α i L(C (∂Ω ),C (∂Ω )) (  ) ( )  1  1 i i  × τϵ   2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]] 1,α i , C (∂Ω )

(82)

where ( ( ) ) 1 i i i ˜ ψ2 =θ2 τ ϵ I + W∂Ω i [ψ ] + ζ 2 ) ) ( ( 1 i i I + W∂Ω i [Ψ [ϵ]] + ζ , + (1 − θ2 ) τ ϵ 2 for some θ2 ∈ ]0, 1[. • Step 6.3: Estimate for T3 . Finally we consider T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·). Adding and subtracting the term ) ( ( ) ) ( 1 1 i 2 2 i i I + W∂Ω i [Ψ [ϵ]] (∂ζ F ) τ ϵ, t, τ ϵ I + W∂Ω i [ψ ](t) + ζ 2 2 on the right hand side of (79) and using the triangular inequality, we obtain ∥T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i ) ( ) ( ( ) )  1 1 i 2 2 i i  ≤ I + W∂Ω i [ψ ] (∂ζ F ) τ ϵ, ·, τ ϵ I + W∂Ω i [ψ ] + ζ 2 2 ( ) ( ( ) )  1 1 i 2 2 i i  − I + W∂Ω i [Ψ [ϵ]] (∂ζ F ) τ ϵ, ·, τ ϵ I + W∂Ω i [ψ ] + ζ  2 2 C 1,α (∂Ω i ) ( ) ( ( ) )  1 1 i 2 2 i i +  2 I + W∂Ω i [Ψ [ϵ]] (∂ζ F ) τ ϵ, ·, τ ϵ 2 I + W∂Ω i [ψ ] + ζ ( ) ( ( ) )  1 1 i 2 2 i i  − I + W∂Ω i [Ψ [ϵ]] (∂ζ F ) τ ϵ, ·, τ ϵ I + W∂Ω i [Ψ [ϵ]] + ζ  . 2 2 C 1,α (∂Ω i )

(83)

By Lemma 6.1 in the Appendix and by the Mean Value Theorem in Banach space (see, e.g., Ambrosetti and Prodi [1, Thm. 1.8]), we can estimate the C 1,α (∂Ω i ) norm of T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) (cf. (79) and (83)) as

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

25

follows: ∥T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i )  ) ) ( (   2 1 i i  [ψ ] + ζ (∂ F ) τ ϵ, ·, τ ϵ I + W ≤ 2 i ∂Ω  1,α i  ζ 2 C ( (∂Ω ) ) ( )  1  1 i 2 i 2  ×  2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]]  1,α i C (∂Ω ) (  )  1  i 2 + 2  2 I + W∂Ω i [Ψ [ϵ]]  1,α i C (∂Ω )  ( ( ) )  1 i i τ ϵ, τ ϵ I + W [ψ ] + ζ N × 2 i ∂Ω  ∂ζ F 2 ) ) ( (  1 i i  I + W∂Ω i [Ψ [ϵ]] + ζ  − N∂ 2 F τ ϵ, τ ϵ ζ 2 C 1,α (∂Ω i )  ( ( ) )  2  1 i i  ≤ 4 (∂ζ F ) τ ϵ, ·, τ ϵ 2 I + W∂Ω i [ψ ] + ζ  1,α i C (  (∂Ω ) ) ( )   1 1 i i  ×  2 I + W∂Ω i [ψ ] − 2 I + W∂Ω i [Ψ [ϵ]] 1,α i ( C (∂Ω ) ) ( )  1  1 i i  ×  2 I + W∂Ω i [ψ ] + 2 I + W∂Ω i [Ψ [ϵ]] 1,α i C (∂Ω )    2 i 2 i  ˜ + 4C3 ∥Ψ [ϵ]∥C 1,α (∂Ω i ) dv N∂ 2 F (τ ϵ, ψ3 ) ζ L(C 1,α (∂Ω i ),C 1,α (∂Ω i ))  ( ) ( )   1 1 i i  I + W∂Ω i [Ψ [ϵ]] × τ ϵ  I + W∂Ω i [ψ ] −  1,α i , 2 2 C (∂Ω )

(84)

where ( ( ) ) 1 ψ˜3i =θ3 τ ϵ I + W∂Ω i [ψ i ] + ζ i 2 ( ( ) ) 1 + (1 − θ3 ) τ ϵ I + W∂Ω i [Ψ i [ϵ]] + ζ i , 2 for some θ3 ∈ ]0, 1[. Let R be as in (70). By the same argument used to prove (69), one verifies the inequalities ∥ψ˜1i ∥C 1,α (∂Ω i ) ≤ R,

∥ψ˜2i ∥C 1,α (∂Ω i ) ≤ R,

∥ψ˜3i ∥C 1,α (∂Ω i ) ≤ R.

(85)

By assumption (9), the partial derivatives ∂ζ ∂ϵ2 F (η, t, ζ), ∂ζ ∂ϵ ∂ζ F (η, t, ζ) and ∂ζ ∂ζ2 F (η, t, ζ) exist for all (η, t, ζ) ∈ ] − ϵ0 , ϵ0 [ × ∂Ω i × R and by Proposition 4.5 and Lemma 6.1(ii) in the Appendix, we obtain ∥dv N∂ϵ2 F (τ η, ψ˜1i )∥L(C 1,α (∂Ω i ),C 1,α (∂Ω i )) ≤ 2∥N∂ζ ∂ϵ2 F (τ η, ψ˜1i )∥C 1,α (∂Ω i ) ∥dv N∂ϵ ∂ζ F (τ η, ψ˜2i )∥L(C 1,α (∂Ω i ),C 1,α (∂Ω i )) ∥dv N∂ 2 F (τ η, ψ˜3i )∥L(C 1,α (∂Ω i ),C 1,α (∂Ω i ))

= 2 ∥∂ζ ∂ϵ2 F (τ η, ·, ψ˜1i (·))∥C 1,α (∂Ω i ) , ≤ 2∥N∂ ∂ ∂ F (τ η, ψ˜i )∥ 1,α i 2

ζ ϵ ζ

C

(∂Ω )

= 2 ∥∂ζ ∂ϵ ∂ζ F (τ η, ·, ψ˜2i (·))∥C 1,α (∂Ω i ) , ≤ ∥N 2 (τ η, ψ˜i )∥ 1,α i ∂ζ ∂ζ F

ζ

=

3

C

(∂Ω )

2∥∂ζ ∂ζ2 F (τ η, ·, ψ˜3i (·))∥C 1,α (∂Ω i )

,

(86)

26

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

for all η ∈ ]0, ϵ∗ [. By Proposition 6.3(ii) in the Appendix, there exists C6 > 0 such that ∥∂ζ ∂ϵ2 F (τ η, ·, ψ˜1i (·))∥C 1,α (∂Ω i ) )2 ( ≤ C6 ∥∂ζ ∂ϵ2 F (τ η, ·, ·)∥C 1,α (∂Ω×[−R,R]) 1 + ∥ψ˜1i ∥C 1,α (∂Ω i ) , ∥∂ζ ∂ϵ ∂ζ F (τ η, ·, ψ˜2i (·))∥C 1,α (∂Ω i ) )2 ( ≤ C6 ∥∂ζ ∂ϵ ∂ζ F (τ η, ·, ·)∥C 1,α (∂Ω i ×[−R,R]) 1 + ∥ψ˜2i ∥C 1,α (∂Ω i ) ,

(87)

∥∂ζ ∂ζ2 F (τ η, ·, ψ˜3i (·))∥C 1,α (∂Ω i ) )2 ( ≤ C6 ∥∂ζ ∂ζ2 F (τ η, ·, ·)∥C 1,α (∂Ω i ×[−R,R]) 1 + ∥ψ˜3i ∥C 1,α (∂Ω i ) , ˜F defined as in Lemma 5.4 (with for all η ∈ ]0, ϵ∗ [. Now, by assumption (9) one deduces that the map N B = F ) is real analytic from ] − ϵ0 , ϵ0 [ × R to C 1,α (∂Ω i ). Then one verifies that also the maps ˜F = N ˜ 2 ∂ϵ2 ∂ζ N ∂ϵ ∂ζ F ,

˜F = N ˜ 2 , ∂ϵ ∂ζ2 N ∂ϵ ∂ F

˜F = N ˜∂ ∂ F , ∂ϵ ∂ζ N ϵ ζ

˜F = N ˜ 2 ∂ζ2 N ∂ F

ζ

˜F = N ˜∂ 3 F , ∂ζ3 N

ζ

are real analytic from ] − ϵ0 , ϵ0 [ × R to C 1,α (∂Ω i ). Hence, Lemma 5.4 (with m = 1) implies that there exists C7 > 0 such that sup η∈[−ϵ∗ ,ϵ∗ ]

sup η∈[−ϵ∗ ,ϵ∗ ]

sup η∈[−ϵ∗ ,ϵ∗ ]

sup η∈[−ϵ∗ ,ϵ∗ ]

sup η∈[−ϵ∗ ,ϵ∗ ]

∥∂ζ ∂ϵ2 F (τ η, ·, ·)∥C 1,α (∂Ω×[−R,R]) ≤ C7 , ∥∂ζ ∂ϵ ∂ζ F (τ η, ·, ·)∥C 1,α (∂Ω×[−R,R]) ≤ C7 , ∥∂ζ ∂ζ2 F (τ η, ·, ·)∥C 1,α (∂Ω×[−R,R]) ≤ C7 ,

(88)

∥∂ϵ ∂ζ F (τ η, ·, ·)∥C 1,α (∂Ω×[ζ i −C3 R,ζ i +C3 R]) ≤ C7 , ∥∂ζ2 F (τ η, ·, ·)∥C 1,α (∂Ω×[ζ i −C3 R,ζ i +C3 R]) ≤ C7 .

Thus, by (85), (86), (87), and (88), and by the membership of ϵ ∈ ]0, ϵ∗ [ and δ ∗ ∈ ]0, 1[, we have ∥dv N∂ϵ2 F (τ ϵ, ψ˜1i )∥L(C 1,α (∂Ω i ),C 1,α (∂Ω i )) ≤ 2C6 C7 (1 + R)2 , ∥dv N∂ϵ ∂ζ F (τ ϵ, ψ˜2i )∥L(C 1,α (∂Ω i ),C 1,α (∂Ω i )) ≤ 2C6 C7 (1 + R)2 , ∥dv N 2 (τ ϵ, ψ˜i )∥ 1,α i 1,α i ≤ 2C6 C7 (1 + R)2 , ∂ζ F

3

L(C

(∂Ω ),C

(89)

(∂Ω ))

uniformly with respect to τ ∈ ]0, 1[. We can now bound the C 1,α norms with respect to the variable t ∈ ∂Ω i of (77), (78), and (79) uniformly with respect to τ ∈ ]0, 1[. Indeed, by (63), (80) and (89), we obtain ∥T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i ) ≤ 2C6 C7 (1 + R)2 δ ∗

(90)

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

for all τ ∈ ]0, 1[. By Proposition 6.3(ii) in the Appendix, by (66) and (88), we obtain  ) ) ( (   (∂ϵ ∂ζ F ) τ ϵ, ·, τ ϵ 1 I + W i [ψ i ] + ζ i  ∂Ω  1,α i  2 C (∂Ω ) ( )2  (  )   1 i i  ≤ C6 C7 1 +  τ ϵ 2 I + W∂Ω i [ψ ] + ζ  1,α i C (∂Ω ) ( ) i 2 ≤ C6 C7 1 + C3 R1 + |ζ | ,  ) ( ( )  2  (∂ζ F ) τ ϵ, ·, τ ϵ 1 I + W i [ψ i ] + ζ i  ∂Ω   1,α i 2 C (∂Ω ) ( )2  (  )   1 i i ≤ C6 C7 1 +  τ ϵ 2 I + W∂Ω i [ψ ] + ζ  1,α i C (∂Ω ) ( ) i 2 ≤ C6 C7 1 + C3 R1 + |ζ | ,

27

(91)

for all τ ∈ ]0, 1[. Hence, in view of (66), (89) and (91) and by (82) and (84) we have ∥T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i ) { } ( )2 ≤ 2C6 C7 1 + C3 R1 + |ζ i | + 4C3 C6 C7 R2 (1 + R)2 δ ∗ , ∥T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)∥C 1,α (∂Ω i ) { } ( )2 ≤ 4C6 C7 1 + C3 R1 + |ζ i | R + 8C32 C6 C7 R22 (1 + R)2 δ ∗ ,

(92)

for all τ ∈ ]0, 1[, where to obtain the inequality for T3 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) we have also used that (  ) ( )  1   I + W i [ψ i ] + 1 I + W i [Ψ i [ϵ]] ∂Ω ∂Ω  2  1,α i ≤ C3 (R1 + R2 ) ≤ R 2 C (∂Ω ) (cf. (70)). Moreover, since the boundedness provided in (90) and (92) is uniform with respect to τ ∈ ]0, 1[, one verifies that the following inequality holds: ∫ 1  {  (1 − τ ) T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) + 2 T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)  0  }  i i +T3 [ϵ, ψ , Ψ [ϵ]](τ, ·) dτ   C 1,α (∂Ω i )

∫ ≤

1

(1 − τ ) ∥ T1 [ϵ, ψ i , Ψ i [ϵ]](τ, ·) + 2 T2 [ϵ, ψ i , Ψ i [ϵ]](τ, ·)

0 i

(93)

i

+ T3 [ϵ, ψ , Ψ [ϵ]](τ, ·) ∥C 1,α (∂Ω i ) dτ )2 ≤ 2C6 C7 (1 + R)2 + 4C6 C7 1 + C3 R1 + |ζ i | + 8C3 C6 C7 R2 (1 + R)2 } ( ) i 2 2 2 2 + 4C6 C7 1 + C3 R1 + |ζ | R + 8C3 C6 C7 R2 (1 + R) δ∗ . {

(

Then, by (76) and (93) we obtain ∥S2 [ϵ, ψ i ] − S2 [ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω i ) { ( )2 ≤ 2C6 C7 (1 + R)2 + 4C6 C7 1 + C3 R1 + |ζ i | + 8C3 C6 C7 R2 (1 + R)2 } ( )2 + 4C6 C7 1 + C3 R1 + |ζ i | R + 8C32 C6 C7 R22 (1 + R)2 δ∗ (also recall that ϵ ∈ ]0, 1[).

(94)

28

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

• Step 7: Conclusion for S. Finally, by (67), (75) and (94), we have ∥S[ϵ, ψ i ] − S[ϵ, Ψ i [ϵ]]∥C 1,α (∂Ω i )×C 0,α (∂Ω i )×C 1,α (∂Ω o ) ≤ C8 δ ∗ , with ( )2 C8 ≡ C4 C5 (1 + Rα ) + 2C6 C7 (1 + R)2 + 4C6 C7 1 + C3 R1 + |ζ i | ( )2 + 8C3 C6 C7 R2 (1 + R)2 + 4C6 C7 1 + C3 R1 + |ζ i | R + 8C32 C6 C7 R22 (1 + R)2 . • Step 8: Estimate for (62) and determination of δ ∗ . By (56) and (57) we conclude that the norm of the difference between (ϕo , ϕi , ζ, ψ i ) and (Φ o [ϵ], Φ i [ϵ], Z[ϵ], Ψ i [ϵ]) in the space C 1,α (∂Ω o ) × C 1,α (∂Ω i )0 × R × C 1,α (∂Ω i ) is less than C1 C8 δ ∗ . Then, by (58) and by the triangular inequality we obtain ∥(ϕo , ϕi , ζ, ψ i ) − (ϕo0 , ϕi0 , ζ0 , ψ0i )∥C 1,α (∂Ω o )×C 1,α (∂Ω i )0 ×R×C 1,α (∂Ω i ) ≤ C1 C8 δ ∗ +

K . 2

Thus, in order to have (ϕo , ϕi , ζ, ψ i ) ∈ B0,K , it suffices to take δ∗ <

K 2C1 C8

in inequality (59). Then, for such choice of δ ∗ , (62) holds and the theorem is proved. □ 5.3. Local uniqueness for the family of solutions As a consequence of Theorem 5.6, we can derive the following local uniqueness result for the family {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ . Corollary 5.7. Let assumptions (7), (8), and (9) hold true. Let ϵ′ ∈]0, ϵ0 [ be as in Theorem 4.6. Let {(uoϵ , uiϵ )}ϵ∈]0,ϵ′ [ be as in Theorem 4.7(v). Let {(vϵo , vϵi )}ϵ∈]0,ϵ′ [ be a family of functions such that (vϵo , vϵi ) ∈ C 1,α (Ω (ϵ)) × C 1,α (ϵΩ i ) is a solution of problem (2) for all ϵ ∈]0, ϵ′ [. If   lim ϵ−1 vϵi (ϵ·) − uiϵ (ϵ·)C 1,α (∂Ω i ) = 0,

ϵ→0+

then there exists ϵ∗ ∈]0, ϵ′ [ such that (vϵo , vϵi ) = (uoϵ , uiϵ )

∀ϵ ∈]0, ϵ∗ [ .

Proof . Let ϵ∗ and δ ∗ be as in Theorem 5.6. By (95) there is ϵ∗ ∈ ]0, ϵ∗ [ such that  i  vϵ (ϵ·) − uiϵ (ϵ·) 1,α i ≤ ϵδ ∗ C (∂Ω ) Then the statement follows from Theorem 5.6. □

∀ϵ ∈ ]0, ϵ∗ [.

(95)

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29

Acknowledgments The research of the R. Molinarolo was supported by HORIZON 2020 RISE project “MATRIXASSAY” under project number 644175. The author gratefully acknowledges the University of Texas at Dallas and the University of Tulsa for the great research environment and the friendly atmosphere provided during the author’s secondment at the University of Texas at Dallas. M. Dalla Riva and P. Musolino are members of the ‘Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni’ (GNAMPA) of the ‘Istituto Nazionale di Alta Matematica’ (INdAM). Appendix In this Appendix, we present some technical facts, which have been exploited in this paper, on product and composition of functions of C 0,α and C 1,α regularity. We start by introducing the following elementary result (cf. Lanza de Cristoforis [2]). Lemma 6.1.

Let n ∈ N \ {0}. Let Ω be a bounded connected open subset of Rn of class C 1,α . Then ∥uv∥C 0,α (∂Ω) ≤ ∥u∥C 0,α (∂Ω) ∥v∥C 0,α (∂Ω)

∀u, v ∈ C 0,α (∂Ω ),

and ∥uv∥C 1,α (∂Ω) ≤ 2 ∥u∥C 1,α (∂Ω) ∥v∥C 1,α (∂Ω)

∀u, v ∈ C 1,α (∂Ω ).

We now present a result on composition of a C m,α function, with m ∈ {0, 1}, with a C 1,α function. Lemma 6.2. Let n, d ∈ N \ {0} and α ∈ ]0, 1]. Let Ω1 be an open bounded convex subset of Rn and Ω2 be an open bounded convex subset of Rd . Let v = (v1 , . . . , vn ) ∈ (C 1,α (Ω2 ))n such that v(Ω2 ) ⊂ Ω1 . Then the following statements hold. (i) If u ∈ C 0,α (Ω1 ), then ( ∥u(v(·))∥C 0,α (Ω2 ) ≤ ∥u∥C 0,α (Ω1 ) 1 + ∥v∥α (C 1,α (Ω

) 2 ))

n

.

(ii) If u ∈ C 1,α (Ω1 ), then ( )2 ∥u(v(·))∥C 1,α (Ω2 ) ≤ (1 + nd)2 ∥u∥C 1,α (Ω1 ) 1 + ∥v∥(C 1,α (Ω2 ))n . Then, by Lemma 6.2, we deduce the following Proposition 6.3. Proposition 6.3. Let n ∈ N \ {0}. Let α ∈]0, 1]. Let Ω be a bounded connected open subset of Rn of class C 1,α . Let R > 0. Then the following hold. (i) There exists c0 > 0 such that ( ) ∥u(·, v(·))∥C 0,α (∂Ω) ≤ c0 ∥u∥C 0,α (∂Ω×[−R,R]) 1 + ∥v∥α 1,α C (∂Ω) for all u ∈ C 0,α (∂Ω × R) and for all v ∈ C 1,α (∂Ω ) such that v(∂Ω ) ⊂ [−R, R]. (ii) There exists c1 > 0 such that ( )2 ∥u(·, v(·))∥C 1,α (∂Ω) ≤ c1 ∥u∥C 1,α (∂Ω×[−R,R]) 1 + ∥v∥C 1,α (∂Ω) for all u ∈ C 1,α (∂Ω × R) and for all v ∈ C 1,α (∂Ω ) such that v(∂Ω ) ⊂ [−R, R].

M. Dalla Riva, R. Molinarolo and P. Musolino / Nonlinear Analysis 191 (2020) 111645

30

Proof . We prove only statement (ii). The proof of statement (i) can be obtained adapting the one of point (ii) and using Lemma 6.2(i) instead of Lemma 6.2(ii). Since ∂Ω is compact and of class C 1,α , a standard argument shows that there are a finite cover of ∂Ω consisting of open subsets U1 , . . . , Uk of ∂Ω and for each j ∈ {1, . . . , k} a C 1,α diffeomorphism γj from Bn−1 (0, 1) to the closure of Uj in ∂Ω . Then, for a fixed j ∈ {1, . . . , k} we define the functions ˜ j : Bn−1 (0, 1) → Bn−1 (0, 1) × R by setting u ˜j : Bn−1 (0, 1) × R → R, v˜j : Bn−1 (0, 1) → R, and w u ˜j (t′ , s) ≡ u(γj (t′ ), s)

∀(t′ , s) ∈ Bn−1 (0, 1) × R,

v˜j (t′ ) ≡ v(γj (t′ ))

∀t′ ∈ Bn−1 (0, 1),

w ˜ j (t′ ) ≡ (t′ , v˜j (t′ ))

∀t′ ∈ Bn−1 (0, 1).

Since v(∂Ω ) ⊂ [−R, R], it follows that w ˜ j (Bn−1 (0, 1)) ⊂ Bn−1 (0, 1) × [−R, R]. Moreover, we can see that there exists dj > 0 such that ∥w ˜ j ∥(C 1,α (B

n−1 (0,1)))

n

≤ dj ∥˜ v j ∥(C 1,α (B

n−1 (0,1))

).

Then Lemma 6.2 implies that there exists cj > 0 such that ∥˜ uj (·, v˜j (·))∥C 1,α (B

n−1 (0,1))

= ∥˜ uj (w ˜ j (·))∥(C 1,α (B

n−1 (0,1)))

≤ cj ∥˜ uj ∥C 1,α (B ≤ cj ∥˜ uj ∥C 1,α (B

n

( n−1 (0,1)×[−R,R])

( n−1 (0,1)×[−R,R])

1 + ∥w ˜ j ∥(C 1,α (B

)2 n n−1 (0,1)))

1 + dj ∥˜ v j ∥C 1,α (B

)2 n−1 (0,1))

(96)

.

Without loss of generality, we can now assume that the norm of C 1,α (∂Ω ) is defined on the atlas {(Uj , γj )}j∈{1,...,k} (cf. Section 2). Then by (96) we have ∥u(·, v(·))∥C 1,α (∂Ω) =

k ∑

∥u(γj (·), v(γj (·)))∥C 1,α (B

n−1 (0,1))

j=1

=

k ∑

∥˜ uj (·, v˜j (·))∥C 1,α (B

(97)

n−1 (0,1))

j=1

≤

k ∑

cj ∥˜ uj ∥C 1,α (B

( n−1 (0,1)×[−R,R])

1 + dj ∥˜ v j ∥C 1,α (B

)2 n−1 (0,1))

j=1

Moreover, ∥˜ uj ∥C 1,α (B

n−1 (0,1)×[−R,R])

≤ ∥u∥C 1,α (∂Ω×[−R,R])

and (

)2 1 + dj ∥˜ v j ∥C 1,α (B (0,1)) n−1 ( 2 ≤ (1 + dj ) 1 + ∥˜ v j ∥C 1,α (B

)2 n−1 (0,1))

( )2 ≤ (1 + dj )2 1 + ∥v∥C 1,α (∂Ω) .

Hence, (97) implies that ∥u(·, v(·))∥C 1,α (∂Ω) )2 ( ≤ k max{c1 (1 + d1 )2 , . . . , ck (1 + dk )2 }∥u∥C 1,α (∂Ω×[−R,R]) 1 + ∥v∥C 1,α (∂Ω) and the proposition is proved. □

.

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