A local uniqueness result for a singularly perturbed ...

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A local uniqueness result for a singularly perturbed periodic nonlinear traction problem Matteo Dalla Riva

∗

Paolo Musolino

†

Abstract: We present a limiting property and a local uniqueness result for converging families of solutions of a singularly perturbed nonlinear traction problem in an unbounded periodic domain with small holes. Keywords: Nonlinear traction problem; Singularly perturbed domain; Linearized elastostatics; Local uniqueness; Integral representation; Elliptic system. 2010 MSC: 35J65; 31B10; 45F15; 74B05.

1

Introduction

In this paper, we exploit an argument based on functional analysis and potential theory to show a limiting property and a local uniqueness result for families of solutions of a singularly perturbed nonlinear traction problem in linearized elasticity. We fix once for all n ∈ N \ {0, 1} ,

q11 , . . . , qnn ∈]0, +∞[ ,

Q ≡ Πnj=1 ]0, qjj [ .

Then we denote by q the n × n diagonal matrix with diagonal entries q11 , . . . , qnn . We also take α ∈]0, 1[ and Ωh ⊆ Rn bounded, open, connected, of class C 1,α , containing the origin 0, and with a connected exterior Rn \ clΩh . Here cl denotes the closure and the letter ‘h’ stands for ‘hole’. The set Ωh will play the role of the shape of the perforation. Moreover, we fix p ∈ Q and 0 ∈]0, +∞[ such that p + clΩh ⊆ Q for all ∈] − 0 , 0 [ . To shorten our notation, we set Ωhp, ≡ p + Ωh and we define the periodically perforated domain S[Ωhp, ]− ≡ Rn \ ∪z∈Zn cl(Ωhp, + qz) ∗ Centro

de Investiga¸c˜ ao e Desenvolvimento em Matem´ atica e Aplica¸c˜ oes (CIDMA), Universidade de Aveiro, Portugal. di Matematica, Universit` a degli Studi di Padova, Italy. The research of M. Dalla Riva was supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (“FCT–Funda¸ca ˜o para a Ciˆ encia e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014. The research was also supported by the Portuguese Foundation for Science and Technology FCT with the research grant SFRH/BPD/64437/2009. P. Musolino is member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The work of M. Dalla Riva and P. Musolino is also supported by “Progetto di Ateneo: Singular perturbation problems for differential operators – CPDA120171/12” - University of Padova. Part of the work was done while P. Musolino was visiting the Centro de Investiga¸c˜ ao e Desenvolvimento em Matem´ atica e Aplica¸co˜ es of the Universidade de Aveiro. P. Musolino wishes to thank the Centro de Investiga¸ca ˜o e Desenvolvimento em Matem´ atica e Aplica¸co˜ es for the kind hospitality. † Dipartimento

1

for all ∈] − 0 , 0 [. A function u defined on clS[Ωhp, ]− is said to be q-periodic if u(x + qz) = u(x)

∀x ∈ clS[Ωhp, ]− , ∀z ∈ Zn .

We now introduce a nonlinear traction boundary value problem in S[Ωhp, ]− . To do so, we denote by T the function from ]1 − (2/n), +∞[×Mn (R) to Mn (R) which takes the pair (ω, A) to T (ω, A) ≡ (ω − 1)(trA)In + (A + At ) . Here Mn (R) denotes the space of n × n matrices with real entries, In denotes the n × n identity matrix, trA and At denote the trace and the transpose matrix of A, respectively. We observe that (ω − 1) plays the role of the ratio between the first and second Lam´e constants and that the classical linearization of the Piola Kirchoff tensor equals the second Lam´e constant times T (ω, ·) (cf., e.g., Kupradze et al. [17]). We also note that div T (ω, Du) = ∆u + ω∇div u , for all regular vector valued functions u. Now let G be a function from ∂Ωh × Rn to Rn , let B ∈ Mn (R), and let ∈]0, 0 [. We introduce the following nonlinear traction problem  in S[Ωhp, ]− ,  div T (ω, Du) = 0 h − u(x + qej ) = u(x) + Bej (1) p, ] , ∀j ∈ {1, . . . , n}, ∀x ∈ clS[Ω  h T (ω, Du(x))νΩhp, (x) = G (x − p)/, u(x) ∀x ∈ ∂Ωp, , where νΩhp, denotes the outward unit normal to ∂Ωhp, and {e1 , . . . , en } denotes the canonical basis of Rn . Because of the presence of a nonlinear term in the third equation of problem (1), we cannot claim in general the existence of a solution. However, we know by [12] that under suitable assumptions there 1,α exists 1 ∈]0, 0 ] such that the boundary value problem in (1) has a solution u(, ·) in Cloc (clS[Ωhp, ]− , Rn ) for all ∈]0, 1 [. Moreover, the family {u(, ·)}∈]0,1 [ is uniquely determined (for small) by its limiting behavior as tends to 0 and the dependence of u(, ·) upon the parameter can be described in terms of real analytic maps of defined in an open neighborhood of 0. In this paper, we are interested in discussing the limiting behavior and the local uniqueness of families of solutions of problem (1), under weaker assumptions than those in [12]. In particular, in Theorem 4.5, we show that if {εj }j∈N is a sequence in ]0, 0 [ converging to 0 and if {uj }j∈N is a family of functions such that uj solves problem (1) for = εj and such that the restrictions to ∂Ωh of the rescaled functions uj (p + εj ·) converge to a function v∗ as j tends to +∞, then v∗ must be equal to a constant vector ξ∗ ∈ Rn and uj converges to ξ∗ + Bq −1 (· − p) uniformly on bounded open subsets of Rn \ (p + qZn ). In Theorem 4.6, instead, we prove that, under suitable assumptions, if {εj }j∈N is a sequence in ]0, 0 [ converging to 0 and if {uj }j∈N , {vj }j∈N are families of functions such that uj and vj solve problem (1) for = εj and such that the restrictions to ∂Ωh of uj (p + εj ·) and of vj (p + εj ·) converge to the same function, then we must have uj = vj for j big enough. We also note that the present article extends to the case of a nonlinear traction problem the results of [9], concerning a nonlinear Robin problem for the Laplace equation. The functional analytic approach adopted in [12] and in the present paper for the investigation of the behavior of the solutions of problem (1) has been previously exploited by Lanza de Cristoforis and the authors to analyze singular perturbation problems for the Laplace operator in [10, 18], for the Lam´e equations in [6, 7, 8], and for the Stokes system in [5]. Concerning problems in an infinite periodically perforated domain, we mention in particular [11, 12, 20, 24]. We note that singularly perturbed boundary value problems have been largely investigated with the methods of asymptotic analysis. As an example, we mention the works of Beretta et al. [2], BonnaillieNo¨el et al. [3], Iguernane et al. [16], Maz’ya et al. [21], Maz’ya et al. [22], Nazarov et al. [25], Nazarov and Sokolowski [26], and Vogelius and Volkov [27]. In particular, in connection with periodic problems, we mention, e.g., Ammari et al. [1]. Moreover, for problems in periodic domains, we mention the method of functional equations and, for example, the works of Castro et al. [4] and Drygas and Mityushev [13] 2

The paper is organized as follows. Section 2 is a section of notation and preliminaries. In Section 3 we provide an integral formulation of problem (1). In Section 4 we prove our main results on the limiting behavior and the local uniqueness of a family of solutions of problem (1).

2

Notation and preliminaries

Let X and Y be normed spaces. We denote by L(X , Y) the space of linear and continuous maps from X to Y, equipped with its usual norm of the uniform convergence on the unit sphere of X . We denote by I the identity operator. The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a real-valued function g, or the inverse of a matrix B, which are denoted g −1 and B −1 , respectively. If B is a matrix, then Bij denotes the (i, j) entry of B. If x ∈ Rn , then xj denotes the j-th coordinate of x and |x| denotes the Euclidean modulus of x. A dot ‘·’ denotes the inner product in Rn . For all R > 0 and all x ∈ Rn we denote by Bn (x, R) the ball {y ∈ Rn : |x − y| < R}. Let O be an open subset of Rn . Let k ∈ N. The space of k times continuously differentiable r real-valued functions on O is denoted by Ck (O). Let r ∈ N \ {0}. Let f ≡ (f1 , . . . , fr ) ∈ C k (O) . Then Df denotes the Jacobian matrix denotes

∂ |η| f η n ∂x1 1 ...∂xη n

∂fs ∂xl

(s,l)∈{1,...,r}×{1,...,n} k

. Let η ≡ (η1 , . . . , ηn ) ∈ Nn , |η| ≡ η1 + · · · + ηn . Then Dη f

. The subspace of C (O) of those functions f whose derivatives Dη f of order |η| ≤ k

can be extended with continuity to clO is denoted C k (clO). Let β ∈]0, 1[. The subspace of C k (clO) whose functions have k-th order derivatives that are uniformly H¨older continuous in clO with exponent β is denoted C k,β (clO) (cf., e.g., Gilbarg and Trudinger [14]). The subspace of C k (clO) of those functions k,β f such that f|cl(O∩Bn (0,R)) ∈ C k,β (cl(O ∩ Bn (0, R))) for all R ∈]0, +∞[ is denoted Cloc (clO). Then k,β k,β k,β n k,β n n n C (clO, R ) denotes (C (clO)) and Cloc (clO, R ) denotes (Cloc (clO)) . If O is a bounded open subset of Rn , then C k,β (clO, Rn ) endowed with its usual norm is well known to be a Banach space. We say that a bounded open subset O of Rn is of class class C k,β , if its closure is a manifold with boundary imbedded in Rn of class C k,β (cf., e.g., Gilbarg and Trudinger [14, §6.2]). If M is a manifold imbedded in Rn of class C k,β with k ≥ 1, then one can define the Schauder spaces also on M by exploiting the local parametrization. In particular, if O is a bounded open set of class C k,β with k ≥ 1, then one can consider the space C l,β (∂O, Rn ) with l ∈ {0, . . . , k} and the trace operator from C l,β (clO, Rn ) to C l,β (∂O, Rn ) is linear and continuous. If SQ is an arbitrary subset of Rn such that clSQ ⊆ Q, then we define [ S[SQ ] ≡ (qz + SQ ) = qZn + SQ , S[SQ ]− ≡ Rn \ clS[SQ ] . z∈Zn

We note that if Rn \ clSQ is connected, then S[SQ ]− is also connected. We now introduce some preliminaries of potential theory. We denote by Sn the function from Rn \ {0} to R defined by 1 ∀x ∈ Rn \ {0}, if n = 2 , sn log |x| Sn (x) ≡ 1 2−n n |x| ∀x ∈ R \ {0}, if n > 2 , (2−n)sn where sn denotes the (n − 1)-dimensional measure of ∂Bn (0, 1). Sn is well-known to be the fundamental solution of the Laplace operator. Let ω ∈]1 − (2/n), +∞[. We denote by Γn,ω the matrix valued function from Rn \ {0} to Mn (R) which takes x to the matrix Γn,ω (x) with (j, k) entry defined by Γkn,ω,j (x) ≡

ω 1 xj xk ω+2 δj,k Sn (x) − 2(ω + 1) 2(ω + 1) sn |x|n

∀(j, k) ∈ {1, . . . , n}2 ,

where δj,k = 1 if j = k, δj,k = 0 if j 6= k. As is well known, Γn,ω is the fundamental solution of the operator L[ω] ≡ ∆ + ω∇div. We find also convenient to set Γkn,ω ≡ Γkn,ω,j j∈{1,...,n} ,

3

which we think as a column vector for all k ∈ {1, . . . , n}. Now let α ∈]0, 1[. Let Ω be a bounded open subset of Rn of class C 1,α . Then we set Z v[ω, µ](x) ≡ Γn,ω (x − y)µ(y) dσy , ∂Ω n

for all x ∈ R and for all µ ≡ (µj )j∈{1,...,n} ∈ C 0,α (∂Ω, Rn ). Here dσ denotes the area element on ∂Ω. As is well known, the elastic single layer potential v[ω, µ] is continuous in the whole of Rn . We set v + [ω, µ] ≡ v[ω, µ]|clΩ and v − [ω, µ] ≡ v[ω, µ]|Rn \Ω . We also find convenient to set Z w∗ [ω, µ](x) ≡

n X

µl (y)T (ω, DΓln,ω (x − y))νΩ (x) dσy

∀x ∈ ∂Ω .

∂Ω l=1

Here νΩ denotes the outward unit normal to ∂Ω. For properties of elastic layer potentials, we refer, e.g., to [6, Appendix A]. We now introduce a periodic analogue of the fundamental solution of L[ω] (cf., e.g., Ammari et al. [1, Lemma 3.2], [12, Thm. 3.1]). Let ω ∈]1 − (2/n), +∞[. We denote by Γqn,ω ≡ (Γq,k n,ω,j )(j,k)∈{1,...,n}2 the matrix of distributions with (j, k) entry defined by " # X ω (q −1 z)j (q −1 z)k 2πi(q−1 z)·x 1 q,k −δj,k + e ∀(j, k) ∈ {1, . . . , n}2 , Γn,ω,j (x) ≡ 2 |Q||q −1 z|2 −1 z|2 4π ω + 1 |q n z∈Z \{0}

where the series converges in the sense of distributions. Then L[ω]Γqn,ω =

X z∈Zn

δqz In −

1 In , |Q|

where δqz denotes the Dirac measure with mass at qz for all z ∈ Zn . Moreover, Γqn,ω is real analytic from Rn \ qZn to Mn (R) and the difference Γqn,ω − Γn,ω can be extended to a real analytic function from q . We find convenient to set (Rn \ qZn ) ∪ {0} to Mn (R) which we denote by Rn,ω q,k q,k q,k Γq,k Rn,ω ≡ Rn,ω,j , n,ω ≡ Γn,ω,j j∈{1,...,n} , j∈{1,...,n} which we think as column vectors for all k ∈ {1, . . . , n}. Let ΩQ be a bounded open subset of Rn of class C 1,α such that clΩQ ⊆ Q. Let µ ∈ C 0,α (∂ΩQ , Rn ). Then we denote by vq [ω, µ] the periodic single layer potential, namely the q-periodic function from Rn to Rn defined by Z vq [ω, µ](x) ≡ Γqn,ω (x − y)µ(y) dσy ∀x ∈ Rn . ∂ΩQ

We also find convenient to set Z wq,∗ [ω, µ](x) ≡

n X

µl (y)T (ω, DΓq,l n,ω (x − y))νΩQ (x) dσy

∀x ∈ ∂ΩQ .

∂ΩQ l=1

Here νΩQ denotes the outward unit normal to ∂ΩQ . If µ ∈ C 0,α (∂ΩQ , Rn ), then the function vq+ [ω, µ] ≡ 1,α vq [ω, µ]|clS[ΩQ ] belongs to Cloc (clS[ΩQ ], Rn ) and the function vq− [ω, µ] ≡ vq [ω, µ]|clS[ΩQ ]− belongs to 1,α Cloc (clS[ΩQ ]− , Rn ). For further properties of vq [ω, ·] and wq,∗ [ω, ·] we refer to [12, Thm. 3.2].

3

An integral equation formulation of the nonlinear traction problem

In this section we provide an integral formulation of problem (1) (cf. [12, §5]). We use the following notation. If G ∈ C 0 (∂Ωh × Rn , Rn ), then we denote by FG the (nonlinear nonautonomous) composition 4

operator from C 0 (∂Ωh , Rn ) to itself which takes v ∈ C 0 (∂Ωh , Rn ) to the function FG [v] from ∂Ωh to Rn defined by FG [v](t) ≡ G(t, v(t)) ∀t ∈ ∂Ωh . Then we consider the following assumption: G ∈ C 0 (∂Ωh × Rn , Rn ) ,

FG maps C 0,α (∂Ωh , Rn ) to itself.

(2)

We also note here that if FG is continuosly Fr´echet differentiable from C 0,α (∂Ωh , Rn ) to itself, then the gradient matrix Du G(·, ·) of G(·, ·) with respect to the variable in Rn exists. Moreover, Du G(·, ξ) ∈ C 0,α (∂Ωh , Mn (R)) for all ξ ∈ Rn , where C 0,α (∂Ωh , Mn (R)) denotes the space of functions of class C 0,α from ∂Ωh to Mn (R) (cf. Lanza de Cristoforis [18, Prop. 6.3]). We now transform problem (1) into an integral equation by means (cf. [12, Prop. 5.2]). R of the following We find convenient to set C 0,α (∂Ωh , Rn )0 ≡ f ∈ C 0,α (∂Ωh , Rn ) : ∂Ωh f dσ = 0 . Proposition 3.1. Let ω ∈]1 − (2/n), +∞[. Let B ∈ Mn (R). Let G be as in assumption (2). Let Λ be the map from ] − 0 , 0 [×C 0,α (∂Ωh , Rn )0 × Rn to C 0,α (∂Ωh , Rn ), defined by Z n X 1 q,l n−1 θl (s)T (ω, DRn,ω ((t − s)))νΩh (t) dσs + T (ω, Bq −1 )νΩh (t) Λ[, θ, ξ](t) ≡ θ(t) + w∗ [ω, θ](t) + 2 ∂Ωh l=1 Z q − G t, v[ω, θ](t) + n−1 Rn,ω ((t − s))θ(s) dσs + Bq −1 t + ξ ∀t ∈ ∂Ωh , ∂Ωh

for all (, θ, ξ) ∈] − 0 , 0 [×C 0,α (∂Ωh , Rn )0 × Rn . If ∈]0, 0 [, then the map u[, ·, ·] from the set of pairs (θ, ξ) ∈ C 0,α (∂Ωh , Rn )0 × Rn that solve the equation Λ[, θ, ξ] = 0

(3)

1,α to the set of functions u ∈ Cloc (clS[Ωhp, ]− , Rn ) which solve problem (1), which takes (θ, ξ) to Z u[, θ, ξ](x) ≡ n−1 Γqn,ω (x − p − s)θ(s) dσs − Bq −1 p + ξ + Bq −1 x ∀x ∈ clS[Ωhp, ]− , ∂Ωh

is a bijection. Hence we are reduced to analyze equation (3). In order to study (1) for small, we first observe that for = 0 we obtain an equation which we address to as the limiting equation and which has the following form 1 θ(t) + w∗ [ω, θ](t) + T (ω, Bq −1 )νΩh (t) − G(t, ξ) = 0 2

∀t ∈ ∂Ωh .

(4)

Then we have the following Proposition, which shows, under suitable assumptions, the solvability of the limiting equation (cf. [12, Prop. 5.3]). Proposition 3.2. Let ω ∈]1 − (2/n), +∞[. Let B ∈ Mn (R). Let G be as in assumption (2). Assume that there exists ξ˜ ∈ Rn such that Z ˜ dσt = 0 . G(t, ξ) ∂Ωh

Then the integral equation 1 ˜ =0 θ(t) + w∗ [ω, θ](t) + T (ω, Bq −1 )νΩh (t) − G(t, ξ) 2

∀t ∈ ∂Ωh

˜ As a consequence, the pair (θ, ˜ ξ) ˜ is a has a unique solution in C 0,α (∂Ωh , Rn )0 , which we denote by θ. 0,α h n n solution in C (∂Ω , R )0 × R of the limiting equation (4). 5

Finally, by a straightforward modification of the proof of [12, Thm. 5.5], we deduce the validity of the following theorem, where we analyze equation (3) around the degenerate value = 0. Theorem 3.3. Let ω ∈]1 − (2/n), +∞[. Let B ∈ Mn (R). Let G be as in assumption (2). Assume that FG is a continuosly Fr´echet differentiable operator from C 0,α (∂Ωh , Rn ) to itself. Assume that there exists ξ˜ ∈ Rn such that Z ˜ dσt = 0 G(t, ξ)

Z det

and

∂Ωh

˜ dσt Du G(t, ξ)

(5)

6= 0.

(6)

∂Ωh

˜ ξ] ˜ =0 Let Λ be as in Proposition 3.1. Let θ˜ be the unique function in C 0,α (∂Ωh , Rn )0 such that Λ[0, θ, ˜ ξ) ˜ in C 0,α (∂Ωh , Rn )0 ×Rn , (cf. Proposition 3.2). Then there exist 1 ∈]0, 0 ], an open neighborhood U of (θ, and a continuously differentiable map (Θ, Ξ) from ] − 1 , 1 [ to U, such that the set of zeros of the map Λ ˜ ξ). ˜ in ] − 1 , 1 [×U coincides with the graph of (Θ, Ξ). In particular, (Θ[0], Ξ[0]) = (θ, Remark 3.4. Let the notation and assumptions of Theorem 3.3 hold. Let u[·, ·, ·] be as in Proposition 3.1. Let u(, x) ≡ u[, Θ[], Ξ[]](x) for all x ∈ clS[Ωhp, ]− and for all ∈]0, 1 [. Then for each ∈]0, 1 [ the function u(, ·) is a solution of problem (1).

4

Converging families of solutions

In this section we investigate some limiting and uniqueness properties of converging families of solutions of problem (1).

4.1

Preliminary results

We first need to study some auxiliary integral operators. In the following lemma, we introduce an operator which we denote by M# . The proof of the lemma can be effected by exploiting classical properties of the elastic layer potentials (see, e.g., [6, Appendix A] and Maz’ya [23, p. 202]). Lemma 4.1. Let ω ∈]1 − (2/n), +∞[. Let M# denote the operator from C 0,α (∂Ωh , Rn )0 × Rn to C 1,α (∂Ωh , Rn ), which takes a pair (θ, ξ) to the function M# [θ, ξ] defined by M# [θ, ξ](t) ≡ v[ω, θ](t) + ξ

∀t ∈ ∂Ωh .

Then M# is a linear homeomorphism. Then, if ∈]0, 0 [, we define the auxiliary integral operator M and we prove its invertibility. Lemma 4.2. Let ω ∈]1−(2/n), +∞[. Let ∈]0, 0 [. Let M denote the operator from C 0,α (∂Ωh , Rn )0 ×Rn to C 1,α (∂Ωh , Rn ) which takes a pair (θ, ξ) to the function M [θ, ξ] defined by Z n−2 q M [θ, ξ](t) ≡ v[ω, θ](t) + Rn,ω ((t − s))θ(s) dσs + ξ ∀t ∈ ∂Ωh . ∂Ωh

Then M is a linear homeomorphism. Proof. We start by proving that M is a Fredholm operator of index 0. We first note that Z n−2 q M [θ, ξ](t) = M# [θ, ξ](t) + Rn,ω ((t − s))θ(s) dσs ∀t ∈ ∂Ωh , ∂Ωh

for all (θ, ξ) ∈ C 0,α (∂Ωh , Rn )0 × Rn . By standard properties of integral operators with real analytic kernels and with no singularity (cf. [19, §4]), we deduce that the linear operator from C 0,α (∂Ωh , Rn )0 to R 2,α h n n−2 C (clΩ , R ) which takes θ to the function Rq ((t − s))θ(s) dσs of the variable t ∈ clΩh is ∂Ωh n,ω 6

continuous. Then by the compactness of the imbedding of C 2,α (clΩh , Rn ) into C 1,α (clΩh , Rn ), and by the continuity of the trace operator from C 1,α (clΩh , Rn ) to C 1,α (∂Ωh , Rn ), and by Lemma 4.1, we deduce that M is a compact perturbation of the linear homeomorphism M# , and thus a Fredholm operator of index 0. Then, by the Fredholm theory, in order to prove that M is a linear homeomorphism, it suffices to show that M is injective. So let (θ, ξ) ∈ C 0,α (∂Ωh , Rn )0 × Rn be such that M [θ, ξ] = 0. Then by the rule of change of variables in integrals, we have 1 · − p x−p ) = vq [ω, θ ](x) + ξ = 0 ∀x ∈ ∂Ωhp, . Then by the periodicity of vq [ω, 1 θ ·−p ] and by a straightforward modification of the argument of [12, Proof of Prop. 4.1], we deduce that M [θ, ξ](

1 · − p vq [ω, θ ](x) + ξ = 0

∀x ∈ clS[Ωhp, ]− .

As a consequence, 1 · − p 1 0 = T (ω, Dvq− [ω, θ ](x))νΩhp, (x) = 2

1 x − p 1 · − p θ ](x) ∀x ∈ ∂Ωhp, . + wq,∗ [ω, θ 2

Then by [12, Prop. 4.4], we deduce that θ = 0 and accordingly ξ = 0. (−1)

We can now show that if {εj }j∈N is a sequence in ]0, 0 [ converging to 0, then Mεj (−1) M# as j → +∞.

converges to

Lemma 4.3. Let ω ∈]1 − (2/n), +∞[. Let {εj }j∈N be a sequence in ]0, 0 [ converging to 0. Then (−1) (−1) in L(C 1,α (∂Ωh , Rn ) , C 0,α (∂Ωh , Rn )0 × Rn ). limj→+∞ Mεj = M# Proof. Let Nj be the operator from C 0,α (∂Ωh , Rn )0 × Rn to C 1,α (∂Ωh , Rn ) which takes (θ, ξ) to Z q Nj [θ, ξ](t) ≡ εn−2 Rn,ω (εj (t − s))θ(s) dσs ∀t ∈ ∂Ωh , ∀j ∈ N. j ∂Ωh

Let UΩh be an open bounded neighborhood of clΩh . Let # be such that (t − s) ∈ (Rn \ qZn ) ∪ {0} for all q in (Rn \ qZn ) ∪ {0} it follows that the map t, s ∈ UΩh and all ∈] − # , # [. By the real analyticity of Rn,ω q which takes (, t, s) to Rn,ω ((t − s)) is real analytic from ] − # , # [×UΩh × UΩh to Mn (R). Hence, there ˜ q from ] − # , # [×UΩh × UΩh to Mn (R) such that Rq ((t −s)) − Rq (0) = exists a real analytic map R n,ω n,ω n,ω q ˜ Rn,ω (, t, s) for all t, s ∈ UΩh and all ∈] − # , # [. Then, by the membership of θ in C 0,α (∂Ωh , Rn )0 , one has Z n−1 ˜ q (εj , t, s)θ(s) dσs Nj [θ, ξ](t) = εj R ∀t ∈ ∂Ωh n,ω ∂Ωh

for all j such that εj ∈]0, # [ and for all (θ, ξ) ∈ C 0,α (∂Ωh , Rn )0 × Rn . Then, by standard properties of integral operators with real analytic kernels and with no singularities (cf. [19, §4]), we can deduce that limj→+∞ Nj = 0 in L(C 0,α (∂Ωh , Rn )0 × Rn , C 1,α (∂Ωh , Rn )). Since Mεj = M# + Nj , it follows that limj→+∞ Mεj = M] in L(C 0,α (∂Ωh , Rn )0 × Rn , C 1,α (∂Ωh , Rn )). Then by the continuity of the map from the open subset of the invertible operators of L(C 0,α (∂Ωh , Rn )0 × Rn , C 1,α (∂Ωh , Rn )) to L(C 1,α (∂Ωh , Rn ) , C 0,α (∂Ωh , Rn )0 × Rn ) which takes an operator to its inverse, one deduces that (−1) (−1) limj→+∞ Mεj = M# (cf. e.g., Hille and Phillips [15, Thms. 4.3.2 and 4.3.3]). 2

4.2

Limiting behavior of a converging family of solutions

We are now ready to investigate in this subsection the limiting behavior of a converging family of solutions of problem (1). To begin with, in the following proposition we consider the limiting behavior of converging families of q-periodic displacement functions. 7

Proposition 4.4. Let ω ∈]1 − (2/n), +∞[. Let {εj }j∈N be a sequence in ]0, 0 [ converging to 0 and let {u#,j }j∈N be a sequence of functions such that for each j ∈ N 1,α u#,j ∈ Cloc (clS[Ωhp,εj ]− , Rn ) , u#,j is q-periodic , and div T (ω, Du#,j ) = 0 in S[Ωhp,εj ]− .

Assume that there exists a function v# ∈ C 1,α (∂Ωh , Rn ) such that lim u#,j (p + εj ·)|∂Ωh = v#

j→+∞

in C 1,α (∂Ωh , Rn ) .

(7)

1,α Then there exists a pair (u# , ξ# ) ∈ Cloc (Rn \ Ωh , Rn ) × Rn such that

v# = u#|∂Ωh + ξ# ,

div T (ω, Du# ) = 0 in Rn \ clΩh ,

and such that supx∈Rn \Ωh |x|n−1+δ2,n |Du# (x)| < ∞ .

supx∈Rn \Ωh |x|n−2+δ2,n |u# (x)| < ∞ , Moreover,

lim u#,j (p + εj ·)|clO\Ωh = u#|clO\Ωh + ξ#

j→+∞

in C 1,α (clO \ Ωh , Rn )

(8)

for all open bounded subsets O of Rn \ clΩh , and ˜ Rn ) in C k (clO,

lim u#,j|clO˜ = ξ#

j→+∞

(9)

˜ of Rn such that clO ˜ ⊆ Rn \ (p + qZn ). for all k ∈ N and for all open bounded subsets O (−1)

(−1)

Proof. Let (θj , ξj ) ≡ Mεj [u#,j (p + εj ·)|∂Ωh ] for all j ∈ N and (θ# , ξ# ) ≡ M# [v# ]. Since the evaluation map from L(C 1,α (∂Ωh , Rn ) , C 0,α (∂Ωh , Rn )0 × Rn ) × C 1,α (∂Ωh , Rn ) to C 0,α (∂Ωh , Rn )0 × Rn , which takes a pair (A, v) to A[v] is bilinear and continuous, the limiting relation (7) and Lemma 4.3 imply that (−1) (10) lim (θj , ξj ) = lim Mε(−1) [u#,j (p + εj ·)|∂Ωh ] = M# [v# ] = (θ# , ξ# ) j j→+∞

j→+∞

in C 0,α (∂Ωh , Rn )0 × Rn . Also, one has Z u#,j (x) = εn−2 Γqn,ω (x − p − εj s)θj (s) dσs + ξj j ∂Ωh

Then one has u#,j (p + εj t) = v[ω, θj ](t) + εn−2 j

Z ∂Ωh

∀x ∈ clS[Ωhp,εj ]− , ∀j ∈ N .

q Rn,ω (εj (t − s))θj (s) dσs + ξj

(11)

(12)

h 0,α for all t ∈ Rn \ ∪z∈Zn (ε−1 (∂Ωh , Rn ) to C 1,α (∂Ωh , Rn ) j qz + clΩ ). By the continuity of the map from C which takes θ to v[ω, θ]|∂Ωh , by standard properties of integral operators with real analytic kernels and R with no singularities (cf. [20, §4]), by condition ∂Ωh θ# dσ = 0, and by (10), one verifies that

lim u#,j (p + εj ·)|∂Ωh = v[ω, θ# ]|∂Ωh + ξ#

j→+∞

in C 1,α (∂Ωh , Rn ) .

Hence, the limiting relation in (7) implies that v# = v[ω, θ# ]|∂Ωh +ξ# . Now the validity of the proposition follows by setting u# (t) ≡ v[ω, θ# ](t) for all t ∈ Rn \ Ωh . Indeed, by classical results for elastic layer R 1,α potentials and by condition ∂Ωh θ# dσ = 0, we have u# ∈ Cloc (Rn \ Ωh , Rn ), div T (ω, Du# ) = 0 in n h R \ clΩ , and supx∈Rn \Ωh |x|n−2+δ2,n |u# (x)| < ∞ ,

supx∈Rn \Ωh |x|n−1+δ2,n |Du# (x)| < ∞ .

Finally, the validity of (8) for all open bounded subsets O of Rn \ clΩh follows by equality (12), by the limiting relation in (10), by the continuity of the map from C 0,α (∂Ωh , Rn ) to C 1,α (clO \ Ωh , Rn ) which 8

takes θ to v[ω, θ]|clO\Ωh , by R standard properties of integral operators with real analytic kernels and with no singularities, and by ∂Ωh θ# dσ = 0. Similarly, the validity of (9) for all k ∈ N and for all open ˜ of Rn such that clO ˜ ⊆ Rn \ (p + qZn ) follows by equality (11), by the limiting relation bounded subsets O in (10), by standard Rproperties of integral operators with real analytic kernels and with no singularities (cf. [20, §4]), and by ∂Ωh θ# dσ = 0. 2 We are now ready to prove the main result of this subsection, where we study the limiting behavior of converging families of solutions of problem (1). Theorem 4.5. Let ω ∈]1 − (2/n), +∞[. Let B ∈ Mn (R). Let G ∈ C 0 (∂Ωh × Rn , Rn ) be such that FG is continuous from C 0,α (∂Ωh , Rn ) to itself. Let {εj }j∈N be a sequence in ]0, 0 [ converging to 0 and let 1,α {uj }j∈N be a sequence of functions such that uj belongs to Cloc (clS[Ωhp,εj ]− , Rn ) and is a solution of (1) with = εj for all j ∈ N. Assume that there exists a function v∗ ∈ C 1,α (∂Ωh , Rn ) such that in C 1,α (∂Ωh , Rn ) .

lim uj (p + εj ·)|∂Ωh = v∗

j→+∞

Then there exists ξ∗ ∈ Rn such that Z

h

v∗ = ξ∗ on ∂Ω ,

G(t, ξ∗ ) dσt = 0 . ∂Ωh

Moreover, lim uj (p + εj ·)|clO\Ωh = ξ∗

j→+∞

in C 1,α (clO \ Ωh , Rn )

for all open bounded subsets O of Rn \ clΩh , and lim uj|clO˜ = ξ∗ + Bq −1 (· − p)

j→+∞

˜ Rn ) in C k (clO,

˜ of Rn such that clO ˜ ⊆ Rn \ (p + qZn ). for all k ∈ N and for all open bounded subsets O Proof. We set u#,j (x) ≡ uj (x) − Bq −1 x

∀x ∈ clS[Ωhp, ]− , ∀j ∈ N ,

v# (x) ≡ v∗ (x) − Bq −1 p

∀x ∈ ∂Ωh .

Then for each j ∈ N 1,α u#,j ∈ Cloc (clS[Ωhp,εj ]− , Rn ) , u#,j is q-periodic , and div T (ω, Du#,j ) = 0 in S[Ωhp,εj ]− ,

and we have v# ∈ C 1,α (∂Ωh , Rn ) ,

lim u#,j (p + εj ·)|∂Ωh = v#

j→+∞

in C 1,α (∂Ωh , Rn ) .

1,α Hence, by Proposition 4.4, there exists a pair (u# , ξ# ) ∈ Cloc (Rn \ Ωh , Rn ) × Rn such that

v# = u#|∂Ωh + ξ# ,

div T (ω, Du# ) = 0

n−2+δ2,n

supx∈Rn \Ωh |x|

n−1+δ2,n

|u# (x)| < ∞ ,

supx∈Rn \Ωh |x|

in Rn \ clΩh ,

(13)

|Du# (x)| < ∞ .

(14)

Moreover, lim u#,j (p + εj ·)|clO\Ωh = u#|clO\Ωh + ξ#

j→+∞

in C 1,α (clO \ Ωh , Rn )

for all open bounded subsets O of Rn \ clΩh , and lim u#,j|clO˜ = ξ#

j→+∞

9

˜ Rn ) in C k (clO,

(15)

˜ of Rn such that clO ˜ ⊆ Rn \ (p + qZn ). Then we observe for all k ∈ N and for all open bounded subsets O that T (ω, Du#,j (p+εj t)+Bq −1 )νΩhp,ε (p+εj t) = G(t, u#,j (p+εj t)+Bq −1 (p+εj t)) j

∀t ∈ ∂Ωh , ∀j ∈ N , (16)

which implies T ω, Dt u#,j (p+εj t) νΩh (t) = −εj T (ω, Bq −1 )νΩh (t)+εj G(t, u#,j (p+εj t)+Bq −1 p+εj Bq −1 t) ∀t ∈ ∂Ωh , (17) for all j ∈ N. Then, by (15), by the continuity of FG from C 0,α (∂Ωh , Rn ) to itself, and by taking the limit as j → +∞ in (17), one obtains ∀t ∈ ∂Ωh ,

T (ω, Du# (t))νΩh (t) = 0

which, together with (13) and (14), implies u# = 0. In particular, in C 1,α (clO \ Ωh , Rn )

lim u#,j (p + εj ·)|clO\Ωh = ξ#

j→+∞

(18)

subsets O of Rn \ clΩh . Furthermore, by (16), by [12, Prop. 4.2], and by the equality Rfor all open bounded −1 T (ω, Bq )νΩhp,ε (x) dσx = 0, one has ∂Ωh p,εj

0=

j

1 εn−1 j

Z T (ω, Du#,j (x) + Bq ∂Ωh p,εj

−1

Z )νΩhp,ε (x) dσx = j

G(t, u#,j (p + εj t) + Bq −1 p + εj Bq −1 t) dσt

∂Ωh

(19) for all j ∈ N. Then, by the continuity of FG from C 0,α (∂Ωh , Rn ) to itself, by the limiting relation in (18), and by letting j → +∞ in (19), one deduces Z G(t, ξ# + Bq −1 p) dσt = 0 . ∂Ωh

Finally, by setting ξ∗ ≡ ξ# + Bq −1 p, the validity of the theorem follows.

4.3

2

A local uniqueness result for converging families of solutions

In this subsection we prove that a converging family of solutions of (1) is essentially unique in a local sense which we clarify in the following theorem. Theorem 4.6. Let ω ∈]1 − (2/n), +∞[. Let B ∈ Mn (R). Let G be as in assumptions (2), (5). Let {εj }j∈N be a sequence in ]0, 0 [ converging to 0. Let {uj }j∈N and {vj }j∈N be sequences of functions such 1,α that uj and vj belong to Cloc (clS[Ωhp,εj ]− , Rn ) and both uj and vj are solutions of (1) with = εj for all j ∈ N. Assume that there exists a function v∗ ∈ C 1,α (∂Ωh , Rn ) such that lim uj (p + εj ·)|∂Ωh = lim vj (p + εj ·)|∂Ωh = v∗

j→+∞

j→+∞

and that

Z det

in C 1,α (∂Ωh , Rn )

(20)

Du G(t, v∗ (t)) dσt

6= 0 .

∂Ωh

Then there exists a natural number j0 ∈ N such that uj = vj for all j ≥ j0 . Proof. We first observe that the family {uj }j∈N and the function v∗ satisfy the conditions in Theorem 4.5. As a consequence, there exists ξ˜ ∈ Rn such that lim uj (p + εj ·)|∂Ωh = ξ˜

j→+∞

10

in C 0,α (∂Ωh , Rn ) .

Then by (20) one also has lim vj (p + εj ·)|∂Ωh = ξ˜

j→+∞

in C 0,α (∂Ωh , Rn ) .

Moreover, we deduce that ξ˜ satisfies assumption (6). By Proposition 3.1, for each j ∈ N there exist and are unique two pairs (θ1,j , ξ1,j ), (θ2,j , ξ2,j ) in C 0,α (∂Ωh , Rn )0 × Rn such that uj (x) = u[εj , θ1,j , ξ1,j ](x) ,

vj (x) = u[εj , θ2,j , ξ2,j ](x) ,

∀x ∈ clS[Ωhp,εj ]− .

(21)

˜ 1 be as in Theorem 3.3. Then to show the validity of the theorem, it will be enough to prove that Let θ, ˜ ξ) ˜ lim (θ1,j , ξ1,j ) = (θ,

in C 0,α (∂Ωh , Rn )0 × Rn ,

(22)

˜ ξ) ˜ lim (θ2,j , ξ2,j ) = (θ,

in C 0,α (∂Ωh , Rn )0 × Rn .

(23)

j→+∞

j→+∞

Indeed, if we denote by U the neighborhood of Theorem 3.3, the limiting relations in (22), (23) imply that there exists j0 ∈ N such that (εj , θ1,j , ξ1,j ), (εj , θ2,j , ξ2,j ) ∈]0, 1 [×U for all j ≥ j0 and thus Theorem 3.3 implies that (θ1,j , ξ1,j ) = (θ2,j , ξ2,j ) = (Θ[εj ], Ξ[εj ]) for all j ≥ j0 , and accordingly the theorem follows by (21). The proof of the limits in (22), (23) follows the lines of [12, Proof of Thm. 7.1] and is accordingly omitted. 2

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