Microscopically weakly singularly perturbed loads for a ...

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Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem. A functional analytic approach. Matteo Dalla Riva and Massimo Lanza de Cristoforis

Abstract i

o

Let Ω and Ω be two bounded open subsets of Rn containing 0. Let G be a (nonlinear) map of ∂Ωi ×Rn to Rn . Let ao be a map of ∂Ωo to the set Mn (R) of n × n matrices with real entries. Let g be a function of ∂Ωo to Rn . Let γ be a positive valued function defined on a right neighborhood of 0 in the real line. Let T be a map of ]1−(2/n), +∞[×Mn (R) to Mn (R). Then we consider the problem  in Ωo \ clΩi ,  div (T (ω, Du)) = 0 1 −T (ω, Du(x))νΩi (x) = γ() Gi (x/, u(x)) ∀x ∈ ∂Ωi ,  T (ω, Du(x))ν o (x) = ao (x)u(x) + g(x) ∀x ∈ ∂Ωo , i

where νΩi and ν o denote the outward unit normal to ∂Ωi and ∂Ωo , respectively, and where > 0 is a small parameter. Here (ω − 1) plays the role of ratio between the first and second Lam´e constants and T (ω, ·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that lim→0 γ −1 ()(log )δ2,n exists in [0, +∞[, we prove that under suitable assumptions the above problem has a family of solutions {u(, ·)}∈]0,0 [ for 0 sufficiently small and we analyze the behaviour of such a family as approaches 0 by an approach which is alternative to those of asymptotic analysis. Here δ2,n denotes the Kronecker symbol.

Key words: Nonlinear traction boundary value problem, singularly perturbed domain, linearized elastostatics, elliptic systems, real analytic continuation in Banach space 2010 MSC : 35J65, 31B10, 45F15, 47H30, 74G99.

1

Introduction.

In a recent paper, we have considered a linearly elastic homogeneous isotropic body with a small hole subject to a traction free boundary condition on the boundary of the hole, and to an external traction depending nonlinearly on the 1

deformation (see [3].) In this paper, we consider the case in which the traction on the boundary of the internal hole depends nonlinearly on the deformation and singularly on the singular perturbation parameter which determines the size of the hole and we consider a linear boundary condition on the external part of the boundary. Of course, one could combine the ideas of the present paper and of [3] and consider nonlinear boundary conditions on all parts of the boundary. We assume that the constitutive relations of our body are expressed by means of the linearized tensor T (ω, ·) defined by T (ω, A) ≡ (ω − 1)(tr A)I + (A + At )

∀A ∈ Mn (R) ,

where ω ∈]1 − (2/n), +∞[ is a parameter such that (ω − 1) plays the role of ratio between the first and second Lam´e constants, Mn (R) denotes the set of n × n matrices with real entries, tr A and At and I denote the trace and the transpose matrix of the matrix A and the identity matrix, respectively. We also note that the classical linearization of the (first) Piola Kirchhoff tensor equals the second Lam´e constant times T (ω, ·). Now we introduce a problem in the case our body has no hole. We assume that the body with no hole occupies an open bounded connected subset Ωo of Rn of class C m,α for some m ∈ N \ {0} and α ∈]0, 1[ and such that 0 ∈ Ωo and such that the exterior of Ωo is also connected. Then we assign a function ao of ∂Ωo to Mn (R), and a function g of ∂Ωo to Rn , and we set Go (t, ξ) ≡ ao (t)ξ + g(t)

∀(t, ξ) ∈ ∂Ωo × Rn .

The function Go plays the role of the reciprocal of the second Lam´e constant times a field of forces applied to the boundary of the body. We know that under certain nondegeneracy assumptions on the function ao , the linear traction boundary value problem div (T (ω, Du)) = 0 in Ωo , (1) o o T (ω, Du(x))ν (x) = a (x)u(x) + g(x) ∀x ∈ ∂Ωo , where ν o denotes the outward unit normal to ∂Ωo , admits a unique solution u ˜ ∈ C m,α (clΩo , Rn ). Next we make a hole in the body Ωo . Namely, we consider another bounded open connected subset Ωi of Rn of class C m,α such that 0 ∈ Ωi and such that the exterior of Ωi is also connected, and we take 0 ∈]0, 1[ such that clΩi ⊆ Ωo for || < 0 , and we consider the perforated domain Ω() ≡ Ωo \ clΩi . Obviously, ∂Ω() = (∂Ωi ) ∪ ∂Ωo . Next we wish to define a boundary value problem in Ω(). To do so, we assign a function Gi of ∂Ωi × Rn to Rn , with the same physical meaning of Go , but for the points of ∂Ωi , and a function γ of ]0, 0 [ to ]0, +∞[, and we consider the following nonlinear problem  in Ω() ,  div (T (ω, Du)) = 0 1 Gi (x/, u(x)) ∀x ∈ ∂Ωi , −T (ω, Du(x))νΩi (x) = γ() (2)  T (ω, Du(x))ν o (x) = ao (x)u(x) + g(x) ∀x ∈ ∂Ωo , 2

where νΩi denotes the outward unit normal to ∂Ωi . The singularity of the load of forces applied to the surface of the hole ∂Ωi is due to the presence of 1 . In this paper, we shall consider the case in which the factor γ() γm ≡ lim γ −1 ()(log )δ2,n ∈ [0, +∞[ , →0

(3)

where δi,j = 1 if i = j, δi,j = 0 if i 6= j. In case γm = 0 ,

(4)

we have what we call the microscopically weakly singular case, and in case γm ∈]0, +∞[ ,

(5)

we have what we call the microscopically singular case. We leave to a future paper, the analysis of the case in which γm is not finite, a case which requires a separate treatment. As we shall see, our analysis yields to consider an auxiliary limiting boundary value problem (see (26)), which consists in a problem for a function ui defined in the exterior of Ωi . Then we show that if such a limiting problem admits a solution which satisfies a certain nondegeneracy condition, then possibly shrinking 0 , problem (2) has a solution u(, ·) ∈ C m,α (clΩ(), Rn ) for all ∈]0, 0 [, which converges to the unique solution u ˜ of (1) in a certain sense, both in case (4) and in case (5). Thus ‘macroscopically’, there is no difference between condition (4) and (5), at least for what concerns the limiting function. Then we turn to consider the microscopic behaviour of u(, ·), i.e., that of the rescaled function u(, t) for t ∈ 1 Ω(), and we show that u(, ·) converges to u ˜(0) plus a term which can be nonzero in case (5), but not in case (4) (see (37).) In this sense we say that case (5) is ‘microscopically’ singular. As we shall see in a future paper, this is not the only reason why such a terminology is appropriate (cf. [4].) Next we pose the following three questions. (j) Let x be a fixed point in clΩo \ {0}. What can be said on the map 7→ u(, x) when is close to 0 and positive? (jj) Let x be a fixed point in Rn \Ωi . What can be said on the map 7→ u(, x) when is close to 0 and positive? (jjj) What can be said on the energy integral Z 1 tr T (ω, Dx u(, x))(Dx u(, x))t dx E(ω, u(, ·)) ≡ 2 Ω() when is close to 0 and positive around = 0? In a sense, question (j) concerns the ‘macroscopic’ behaviour, whereas question (jj) concerns the ‘microscopic’ behaviour of u(, ·). Questions of this type have long been investigated for linear problems with the methods of Asymptotic Analysis and of the Calculus of the Variations. Here, we mention Dal 3

Maso and Murat [5], Kozlov, Maz’ya and Movchan [10], Maz’ya, Nazarov and Plamenewskii [16], Ozawa [17], Ward and Keller [20]. We also mention the seminal paper of Ball [2] on nonlinear elastic cavitation. For more comments, see also [3]. Here we wish to represent the maps of (j), (jj) or of (jjj) in terms of real analytic maps and in terms of possibly singular at 0, but known functions of (such as −1 , log , etc..) We observe that our approach does have certain advantages. Indeed, if for example we know that the map in (jjj) equals for > 0 a real analytic function defined in a whole neighborhood of = 0, then we know that such a map can be expanded in power series for small. As we shall see, this is the case for example if γ() = 1 and n ≥ 3. Such a project has been carried out for a number of problems. For corresponding references, we refer to [12], [3]. In particular, in [12], a nonlinear Robin problem for the Laplace operator on a domain as Ω() has been considered. Here we generalize the techniques of [12] to the case of the elliptic system of linearized elasticity and to the case in which the boundary data are also perturbed singularly.

2

Preliminaries and Notation

We denote the norm on a (real) normed space X by k · kX . Let X and Y be normed spaces. We endow the product space X × Y with the norm defined by k(x, y)kX ×Y ≡ kxkX + kykY ∀(x, y) ∈ X × Y, while we use the Euclidean norm for Rn . For standard definitions of Calculus in normed spaces, we refer to Prodi and Ambrosetti [19]. The symbol N denotes the set of natural numbers including 0. Throughout the paper, n is an element of N \ {0, 1} . The inverse function of an invertible function f is denoted f (−1) , as opposed to the reciprocal of a complex-valued function g, or the inverse of a matrix A, which are denoted g −1 and A−1 , respectively. A dot ‘·’ denotes the inner product in Rn , or the matrix product between matrices with real entries. Let D ⊆ Rn . Then cl D denotes the closure of D and ∂D denotes the boundary of D. For all R > 0, x ∈ Rn , xj denotes the j-th coordinate of x, |x| denotes the Euclidean modulus of x in Rn , and Bn (x, R) denotes the ball {y ∈ Rn : |x − y| < R} and we set Bn ≡ Bn (0, 1). Let Ω be an open subset of Rn . The space of m times continuously differentiable real-valued functions on Ω is n denoted by C m (Ω, R), or more simply by C m (Ω). Let f ∈ (C m (Ω)) . The s-th component of f is denoted fs , and Df (or ∇f ) denotes the gradient matrix ∂fs ∂xl

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

s,l=1,...,n |η| f denotes ∂xη∂1 ...∂x ηn n 1

. 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 cl Ω is denoted C m (cl Ω). The subspace of C m (cl Ω) whose functions have m-th order derivatives that are H¨ older continuous with exponent α ∈]0, 1] is denoted 4

C m,α (cl Ω), (cf. e.g. Gilbarg and Trudinger [8].) The subspace of C m (cl Ω) of those functions f such that f|cl(Ω∩Bn (0,R)) ∈ C m,α (cl(Ω ∩ Bn (0, R))) for all m,α R ∈]0, +∞[ is denoted Cloc (cl Ω). Let D ⊆ Rn . Then C m,α (cl Ω, D) denotes n m,α {f ∈ (C (cl Ω)) : f (cl Ω) ⊆ D}. Now let Ω be a bounded open subset of Rn . P m Then C (cl Ω) endowed with the norm kf kC m (cl Ω) ≡ |η|≤m supcl Ω |Dη f | is a 0,α Banach n space. If f ∈ C (cl Ω), then o its H¨older constant |f : Ω|α is defined as |f (x)−f (y)| |x−y|α

: x, y ∈ cl Ω, x 6= y . The space C m,α (cl Ω), equipped with its P usual norm kf kC m,α (cl Ω) = kf kC m (cl Ω) + |η|=m |Dη f : Ω|α , is well-known to be a Banach space. We say that a bounded open subset of Rn is of class C m or of class C m,α , if it is a manifold with boundary imbedded in Rn of class C m or C m,α , respectively (cf. e.g., Gilbarg and Trudinger [8, §6.2].) For standard properties of the functions of class C m,α both on a domain of Rn or on a manifold imbedded in Rn we refer to Gilbarg and Trudinger [8] (see also [14, §2, Lem. 3.1, 4.26, Thm. 4.28], Lanza and Rossi [15, §2].) We retain the standard notation of Lp spaces and of corresponding norms. We note that throughout the paper ‘analytic’ means ‘real analytic’. For the definition and properties of analytic operators, we refer to Prodi and Ambrosetti [19, p. 89]. We denote by Sn the function of Rn \ {0} to R defined by 1 ∀ξ ∈ Rn \ {0}, if n = 2 , 2π log |ξ| Sn (ξ) ≡ 1 2−n n |ξ| ∀ξ ∈ R \ {0}, if n > 2 , (2−n)sn sup

where sn denotes the (n − 1) dimensional measure of ∂Bn . Sn is well-known to be the fundamental solution of the Laplace operator. We denote by Γn (·, ·) the matrix valued function of (R \ {−1}) × (Rn \ {0}) to Mn (R) which takes a pair (ω, ξ) to the matrix Γn (ω, ξ) defined by Γjn,i (ω, ξ) ≡

ω+2 ω 1 ξi ξj δi,j Sn (ξ) − , 2(ω + 1) 2(ω + 1) sn |ξ|n

for all i, j ∈ {1, . . . , n}. As is well known, Γn (ω, ξ) is the fundamental solution of the operator L[ω] ≡ ∆ + ω∇div . We note that the classical operator of linearized homogeneous and isotropic elastostatics equals L[ω] times the reciprocal of the second constant of Lam´e, and that L[ω]u = div T (ω, Du) for all regular vector valued functions u, and that the classical fundamental solution of the operator of linearized homogeneous and isotropic elastostatics equals Γn (ω, ξ) times the reciprocal of the second constant of Lam´e. We find also convenient to set Γjn (·, ·) ≡ (Γjn,i (·, ·))i=1,...,n , which we think of as column vectors, for all j = 1, . . . , n. Let α ∈]0, 1[. Let Ω be an open bounded connected subset of Rn of class C 1,α . We shall denote by νΩ the outward unit normal to ∂Ω. We also set Ω− ≡ Rn \ clΩ . 5

Let ω ∈]1 − (2/n), +∞[. Then we set Z v[ω, µ](x) ≡ Γn (ω, x − y)µ(y) dσy , ∂Ω Z t i w[ω, µ](x) ≡ − µ (y)T (ω, Dξ Γn (ω, x − y))νΩ (y) dσy ∂Ω

, i=1,...,n

for all x ∈ Rn and for all µ ≡ (µj )j=1,...,n ∈ L2 (∂Ω, Rn ). As is well known, if µ ∈ C 0,α (∂Ω, Rn ), then v[ω, µ] is continuous in the whole of Rn , and we set v − [ω, µ] ≡ v[ω, µ]|clΩ− .

v + [ω, µ] ≡ v[ω, µ]|clΩ

Also if µ ∈ C 0,α (∂Ω, Rn ), then w[ω, µ]|Ω admits a unique continuous extension to clΩ, which we denote by w+ [ω, µ], and w[ω, µ]|Ω− admits a unique continuous extension to clΩ− , which we denote by w− [ω, µ]. We now shortly review some facts on the linear traction problem, which we need in the sequel. Let a be a continuous map of ∂Ω to Mn (R) satisfying the following assumptions. The determinant det a(·) of a(·) does not vanish identically in ∂Ω , t

ξ a(x)ξ ≥ 0

n

∀x ∈ ∂Ω, ∀ξ ∈ R .

(6) (7)

Then we have the following known result. Proposition 2.1. Let ω ∈]1 − (2/n), +∞[. Let Ω be a bounded open connected subset of Rn of class C 1 . Let a ∈ C 0 (∂Ω, Mn (R)) satisfy conditions (6) and (7). Let g ∈ C 0 (∂Ω, Rn ). Then the following statements hold. (i) There exists at most one function u ∈ C 1 (clΩ, Rn ) ∩ C 2 (Ω, Rn ) such that div (T (ω, Du)) = 0 in Ω , (8) T (ω, Du)νΩ + au = g on ∂Ω . (ii) If Ω− is connected, then there exists at most one u ∈ C 1 (clΩ− , Rn ) ∩ C 2 (Ω− , Rn ) such that  div (T (ω, Du)) = 0 in Ω− ,    −T (ω, Du)νΩ + au = g on ∂Ω , (9)  supx∈Ω− |x|n−2 |u(x)| < ∞ ,   supx∈Ω− |x|n−1+δ2,n |Du(x)| < ∞ . Proof. The proof of statement (i) is based on a classical energy argument. We refer for example to Kupradze et al. [11, Ch. III, §1, Thm. 1.6] (see also [3, Prop. 2.1].) We now prove statement (ii). Let u ∈ C 1 (clΩ− , Rn ) ∩ C 2 (Ω− , Rn ) solve problem (9) with g = 0. Let R > 0 be such that clΩ ⊆ Bn (0, R). By

6

applying the Divergence Theorem to u on Bn (0, R) \ clΩ, we obtain that Z tr T (ω, Du)(Du)t dx Bn (0,R)\clΩ Z Z =− ut T (ω, Du)νΩ dσ + ut T (ω, Du)νBn (0,R) dσ ∂Ω ∂Bn (0,R) Z Z =− ut au dσ + ut T (ω, Du)νBn (0,R) dσ . ∂Ω

∂Bn (0,R)

Now by taking the limit as R tends to infinity and by exploiting condition (7) and the last two inequalities of (9), we obtain that Z Z tr T (ω, Du)(Du)t dx = − ut au dσ ≤ 0 . Ω−

∂Ω

Since ω ∈]1 − (2/n), +∞[, a standard computation shows that there exists c > 0 such that tr T (ω, A)At

≥ c|A + At |2

∀A ∈ Mn (R) ,

(cf. e.g., [3, Lem. 6.1].) Hence, |Du + Dt u| must equal zero almost everywhere in Ω− . Then as is well known, there exist a skew-symmetric matrix A ∈ Mn (R) and b ∈ Rn such that u(x) = Ax + b for all x ∈ Ω− . Then by invoking the decay conditions of (9), we can easily see that A = 0. Hence, u = b on Ω− . Then by the boundary conditions and by condition (6), we deduce that b = 0. 2 As customary, we associate to problem (8) and to problem (9) an integral equation. For each a ∈ C 0 (∂Ω, Mn (R)), and ω ∈]1 − (2/n), +∞[, and µ ∈ L2 (∂Ω, Rn ), we set 1 J± a [ω, µ] ≡ ∓ µ + v∗ [ω, µ] ± av[ω, µ] 2 ˜ ± [ω, c, µ] ≡ ∓ 1 µ + v∗ [ω, µ] ± av[ω, µ] ± ac J a 2

on ∂Ω , on ∂Ω ,

where Z v∗ [ω, µ](x) ≡

n X

µl (y)T (ω, Dξ Γln (ω, x − y))νΩ (x) dσy

∀x ∈ ∂Ω .

∂Ω l=1

Also, if X is a vector subspace of L1 (∂Ω, Rn ), we find convenient to set Z X0 ≡ f ∈ X : f dσ = 0 . ∂Ω

Then we have the following classical result. Theorem 2.2. Let α ∈]0, 1[, ω ∈]1 − (2/n), +∞[, m ∈ N \ {0}. Let Ω be an open bounded connected subset of Rn of class C m,α . Let the function a ∈ C m−1,α (∂Ω, Mn (R)) satisfy conditions (6), (7). Then the following statements hold. 7

R ˜ + [ω, c, µ]) is a (i) The map which takes a pair (c, µ) to the pair ( ∂Ω µ dσ, J a n m−1,α n homeomorphism of R × C (∂Ω, R ) onto itself. In particular the ˜ + [ω, c, µ] is a homeomorphism of Rn × map which takes a pair (c, µ) to J a m−1,α n m−1,α C (∂Ω, R )0 onto C (∂Ω, Rn ). (ii) Let g ∈ C m−1,α (∂Ω, Rn ). Then the boundary value problem (8) admits a unique solution u ∈ C m,α (clΩ, Rn ), and u = v[ω, µ] + c, where (c, µ) ∈ Rn × C m−1,α (∂Ω, Rn )0 is the unique solution of equation ˜ + [ω, c, µ] = g J a

on ∂Ω .

(iii) Let n R= 2. Let Ω− be connected. The map which takes a pair (c, µ) to the ˜ − [ω, c, µ]) is a homeomorphism of R2 × C m−1,α (∂Ω, R2 ) pair ( ∂Ω µ dσ, J a ˜ − [ω, c, µ] is onto itself. In particular the map which takes a pair (c, µ) to J a a homeomorphism of R2 × C m−1,α (∂Ω, R2 )0 onto C m−1,α (∂Ω, R2 ). (iv) Let n = 2. Let Ω− be connected. Then the boundary value problem (9) m,α admits a unique solution u ∈ Cloc (clΩ− , R2 ), and u = v[ω, µ] + c, where (c, µ) ∈ R2 × C m−1,α (∂Ω, R2 )0 is the unique solution of equation ˜ − [ω, c, µ] = −g J a

on ∂Ω .

(v) Let n ≥ 3. Let Ω− be connected. Then the operator J− a [ω, ·] is a linear homeomorphism of C m−1,α (∂Ω, Rn ) onto itself. (vi) Let n ≥ 3. Let Ω− be connected. Then the boundary value problem (9) has m,α one and only one solution u ∈ Cloc (clΩ− , Rn ), and u = v[ω, µ], where m−1,α n µ∈C (∂Ω, R ) is the unique solution of equation J− a [ω, µ] = −g

on ∂Ω .

Proof. Statements (i) and (ii) should be considered ‘folklore’ and can be proved by a standard argument. For a proof, we refer for example to [3, Thm. 2.2]. A proof of statements (iii)–(vi) can be effected by following the lines of those of statements (i), (ii) and by replacing the use of Proposition 2.1 (i) with that of Proposition 2.1 (ii). 2 We shall also need the following well known result concerning the Neumann problem. Theorem 2.3. Let α ∈]0, 1[, ω ∈]1 − (2/n), +∞[. Let m ∈ N \ {0}. Let Ω be an open bounded connected subset of Rn of class C m,α such that Ω− is connected. Then the following statements hold. (i) The map of C m−1,α (∂Ω, Rn ) to itself which takes η to linear homeomorphism.

8

1 2η

+ v∗ [ω, η] is a

(ii) Let g ∈ C m−1,α (∂Ω, Rn ). Then the boundary value problem  in Ω− ,   div (T (ω, Du)) = 0  −T (ω, Du)νΩ = g on ∂Ω , n−2+δ2,n ] sup |u (x)| < ∞ , − |x|  x∈Ω   supx∈Ω− |x|n−1+δ2,n |Du] (x)| < ∞ , where

Z

]

u (x) ≡ u(x) + δ2,n Γn (ω, x)

g dσ

(10)

∀x ∈ Ω− ,

∂Ω m,α Cloc (clΩ− , Rn ),

has one and only one solution u ∈ and u = v − [ω, µ], m−1,α n where η ∈ C (∂Ω, R ) is the unique solution of the equation 1 η + v∗ [ω, η] = −g 2

on ∂Ω

(11)

Proof. For a proof of statement (i), we refer to Kupradze et al. [11, Ch. VI, §5.1] (see also [3, Rmk. 6.8].) We now prove statement (ii). Let η ∈ C m−1,α (∂Ω, Rn ) be as in (11). Then by classical properties of elastic layer potentials, the function u ≡ v − [ω, µ] satisfies the first and second equation of (10). We now prove that u satisfies the last two conditions of (10). Case n ≥ 3 follows immediately by the definition of Γn . We now consider case n = 2. By integrating on ∂Ω the jump relations, we deduce that Z Z T (ω, Du)νΩ dσ = η dσ , ∂Ω

∂Ω

(cf. e.g., [3, right above Thm. 6.6].) Then we note that Z ] u (x) ≡ u(x) + Γ2 (ω, x) g dσ ∂Ω Z Z = Γ2 (ω, x − y)η(y) dσ − Γ2 (ω, x) T (ω, Du)νΩ dσ ∂Ω ∂Ω Z Z = Γ2 (ω, x − y)η(y) dσ − Γ2 (ω, x) η dσ ∂Ω Z∂Ω = (Γ2 (ω, x − y) − Γ2 (ω, x))η(y) dσy .

(12)

∂Ω

Now let R ∈]0, +∞[ be such that clΩ ⊆ B2 (0, R). Since the functions |x|(Γ2 (ω, x − y) − Γ2 (ω, x)) ,

|x|2 (Dx Γ2 (ω, x − y) − Dx Γ2 (ω, x)) ,

(13)

are bounded for (x, y) ∈ (R2 \clB2 (0, R))×∂Ω, we deduce that both the functions |x||u] (x)| and |x|2 |Du] (x)| are bounded in R2 \ clB2 (0, R) and thus on Ω− . Hence, u satisfies problem (10). We now prove uniqueness for (10). To do so, we assume that g = 0, and we show that a solution u of (10) must necessarily vanish identically. Let R ∈ 9

]0, +∞[ be such that clΩ ⊆ Bn (0, R). By integrating tr (T (ω, Du)(Du)t ) on Bn (0, R) \ clΩ, and by expoiting the Divergence Theorem, and the boundary condition on ∂Ω, and by letting R tend to infinity, we deduce that Z tr (T (ω, Du)(Du)t ) dx = 0 . Ω−

Then by arguing exactly as in the proof of Proposition 2.1, we deduce that there exist a skew-symmetric matrix A and b ∈ Rn such that u(x) = Ax + b for all x ∈ Ω− . Since u has limit 0 as x tends to infinity, we conclude that A and b must both vanish. 2 For each G ∈ C 0 (∂Ω × Rn , Rn ), we denote by FG the superposition operator of C 0 (∂Ω, Rn ) to itself which maps v ∈ C 0 (∂Ω, Rn ) to the function FG [v] ∈ C 0 (∂Ω, Rn ) defined by FG [v](t) ≡ G(t, v(t))

∀t ∈ ∂Ω .

We now transform our nonlinear boundary value problem into a problem for integral equations by means of the following proposition (see [3, Prop. 2.3].) Proposition 2.4. Let α ∈]0, 1[, ω ∈]1 − (2/n), +∞[. Let m ∈ N \ {0}. Let Ω be an open bounded connected subset of Rn of class C m,α . Let G ∈ C 0 (∂Ω×Rn , Rn ) be such that FG maps C m−1,α (∂Ω, Rn ) to itself. Then the map of the set of pairs (c, µ) ∈ Rn × C m−1,α (∂Ω, Rn )0 which satisfy the problem ˜ + [ω, c, µ] = FG [v[ω, µ]|∂Ω + c] J 0

(14)

to the set of u ∈ C m,α (clΩ, Rn ) which solve the problem div (T (ω, Du)) = 0 in Ω , T (ω, Du)νΩ = FG [u|∂Ω ] on ∂Ω , which takes (c, µ) to the function v + [ω, µ] + c is a bijection.

3

Formulation of the problem in terms of integral equations, and existence of the solution u(, ·)

We now provide a formulation of problem (2) in terms of integral equations. We shall consider the following assumptions for some α ∈]0, 1[ and for some m ∈ N \ {0}. Let Ω be a bounded open connected subset of Rn of class C m,α .(15) Let Rn \ clΩ be connected. Let 0 ∈ Ω . 10

Now let Ωi , Ωo be as in (15). Then there exists 0 ∈]0, 1[ such that clΩi ⊆ Ωo , ∀ ∈] − 0 , 0 [ .

(16)

A simple topological argument shows that Ω() ≡ Ωo \ clΩi is connected, and that Rn \ clΩ() has exactly the two connected components Ωi and Rn \ clΩo , and that ∂Ω() = (∂Ωi ) ∪ ∂Ωo , for all ∈] − 0 , 0 [\{0}. For brevity, we set ν o ≡ νΩo .

ν i ≡ νΩi Obviously,

νΩ() (x) = −ν i (x/) sgn()

∀x ∈ ∂Ωi ,

o

∀x ∈ ∂Ωo ,

νΩ() (x) = ν (x)

for all ∈] − 0 , 0 [\{0}, where sgn() = 1 if > 0, sgn() = −1 if < 0. Then we shall consider the following assumptions Gi ∈ C 0 (∂Ωi × Rn , Rn ) , FGi maps C g∈C

m−1,α

m−1,α o

i

(17) n

(∂Ω , R ) to itself , n

o

(∂Ω , R ), a ∈ C

m−1,α

(18) o

(∂Ω , Mn (R)),

(19)

and we set Go (t, ξ) ≡ g(t) + ao (t)ξ

∀(t, ξ) ∈ ∂Ωo × Rn ,

(20)

Incidentally, we note that condition (18) does not imply condition (17) (cf. e.g., Appell and Zabrejko [1, p. 190].) Moreover, condition (18) is certainly satisfied if Gi is for example of class C m . If Gi ∈ C 0 (∂Ωi × Rn ), Go ∈ C 0 (∂Ωo × Rn ), we denote by G the function of ∂Ω() × Rn to Rn defined by G(s, ξ) ≡ Go (s, ξ) i

G(s, ξ) ≡ G (s/, ξ)

if (s, ξ) ∈ ∂Ωo × Rn , i

(21)

n

if (s, ξ) ∈ ∂Ω × R .

We now convert our boundary value problems (1) and (2) into integral equations. We could exploit Proposition 2.4. However, we note that the corresponding representation formulas include integrations on the -dependent set ∂Ω(). In order to get rid of such dependence, we introduce the following theorem in which we properly rescale the restriction of the unknown function to ∂Ωi . We note that the transformation we operate (cf. (23)) differs considerably from that we have operated for the treatment of the nonlinear conditions on ∂Ωo of [3]. We find convenient to introduce the following notation. We set Xm,α ≡ C m−1,α (∂Ωi , Rn ) × C m−1,α (∂Ωo , Rn ) ,

11

and we introduce the map M = (M1 , M2 , M3 ) of ] − 0 , 0 [×Rn+3 × Xm,α to Rn × Xm,α defined by Z Z M1 [, 1 , 2 , 3 , c, η, ρ] ≡ 2 η dσ + ρ dσ , ∂Ωi

∂Ωo

M2 [, 1 , 2 , 3 , c, η, ρ](t) Z n X 1 ≡ η(t) + v∗ [ω, η](t) + 1 ρl (s)T (ω, Dξ Γln (ω, t − s))ν i (t) dσs 2 ∂Ωo l=1 i +3 G t, (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 η dσ + v[ω, ρ](t) + c ∀t ∈ ∂Ωi , + 4π ω + 1 ∂Ωi M3 [, 1 , 2 , 3 , c, η, ρ](t) 1 ≡ − ρ(t) + v∗ [ω, ρ](t) 2 Z n X ηl (s)T (ω, Dξ Γln (ω, t − s))ν o (t) dσs +2 ∂Ωi l=1 o

Z

−g(t) − a (t) · 2

Γn (ω, t − s)η(s) dσs + v[ω, ρ](t) + c

∀t ∈ ∂Ωo ,

∂Ωi

for all (, 1 , 2 , 3 , c, η, ρ) ∈] − 0 , 0 [×Rn+3 × Xm,α . The map M will play an important role in the next two statements. Theorem 3.1. Let α ∈]0, 1[, ω ∈]1 − (2/n), +∞[, m ∈ N \ {0}. Let Ωi , Ωo be as in (15). Let 0 be as in (16). Let Gi , Go be as in (17), (18), (19), (20). Let γ(·) be a map of ]0, 0 [ to ]0, +∞[. Let ∈]0, 0 [. The map u [·, ·, ·] of the set of solutions (c, η, ρ) ∈ Rn × Xm,α of equation (log )δ2,n n−2 δ2,n , , c, η, ρ = 0 , (22) M , (log ) , (log )δ2,n γ() to the set of solutions u ∈ C m,α (clΩ(), Rn ) of (2) which takes (c, η, ρ) to v + [ω, µ] + c, where µ(x) ≡ ρ(x)

if x ∈ ∂Ωo ,

µ(x) ≡

1 η(x/) (log )δ2,n

if x ∈ ∂Ωi ,

(23)

is a bijection. Proof. Let > 0. A simple computation based on the rule of change of variables in integrals over ∂Ωi shows that (c, η, ρ) solves equation (22) if and only if the pair (c, µ) solves the integral equation (14) with Ω = Ω() and G as in (21). Thus the statement follows by Proposition 2.4. 2

12

Theorem 3.1 reduces the analysis of problem (2), which has been considered only for ∈]0, 0 [ to that of equation M = 0. However equation M = 0, contrary to problem (2) makes sense also for = 0. Then we state the following Theorem which clarifies what equation M = 0 means if = 0. Theorem 3.2. Let the same assumptions of Theorem 3.1 hold. Let γm ∈ [0, +∞[. The triple (c, η, ρ) ∈ Rn × Xm,α satisfies the equation M [0, 0, 0, γm , c, η, ρ] = 0

(24)

if and only if both the following conditions are satisfied (j) (c, ρ) coincides with the unique pair (˜ c, ρ˜) in Rn × C m−1,α (∂Ωo , Rn )0 such + that v [ω, ρ˜] + c˜ is the only solution u ˜ in C m,α (clΩo ) of (1). (jj) η satisfies equation M2 [0, 0, 0, γm , c˜, η, ρ˜] = 0 .

(25)

The map ui0 [·] of the set of solutions η ∈ C m−1,α (∂Ωi , Rn ) of equation (25) m,α to the set of solutions ui ∈ Cloc (Rn \ Ωi , Rn ) of the limiting boundary value problem  i )) = 0 in Rn \ clΩi ,   div (T (ω, Du  i i    −T (ω, Du (x))ν (x)    δ2,n ω+2 R i i i T (ω, Du )ν dσ + u ˜ (0) = γm G x, (1 − δ2,n )ui (x) + 4π ω+1 ∂Ωi (26)  i  ∀x ∈ ∂Ω ,     supx∈Ωi− |x|n−2+δ2,n |u],i (x)| < ∞ ,    supx∈Ωi− |x|n−1+δ2,n |Du],i (x)| < ∞ , where u],i (x) ≡ ui (x) − δ2,n Γn (ω, x)

Z

T (ω, Dui )ν i dσ

∀x ∈ Rn \ Ωi ,

∂Ωi

and which takes η to the function ui0 [η] ≡ v − [ω, η] , is a bijection. In particular, if γm = 0, then η = 0 and ui0 [η] = 0. Proof. The equivalence of conditions (j) and (jj) and of equation (24) follows immediately by Theorem 2.2 (i), (ii). Now we prove the second part of the statement. Let η ∈ C m−1,α (∂Ωi , Rn ) satisfy equation (25). Then ui0 [η] ∈ m,α Cloc (Rn \ Ωi , Rn )R and by standard jump properties of single elastic layer poR tentials we have ∂Ωi T (ω, Dui0 [η])ν i dσ = ∂Ωi η dσ (cf. e.g., [3, right above Thm. 6.6].) Hence, ui0 [η] satisfies the first two conditions of (26). Then by the boundedness properties of the functions in (13), and by the same computations

13

of the last part of (12), we deduce that ui0 [η] satisfies the last two inequalities m,α of (26). Conversely, let ui ∈ Cloc (Rn \ Ωi , Rn ) satisfy (26). Then we set Z δ2,n ω + 2 g i (x) ≡ γm Gi x, (1 − δ2,n )ui (x) + T (ω, Dui )ν i dσ + u ˜(0) 4π ω + 1 ∂Ωi for all x ∈ ∂Ωi . Since Ωi and Rn \ clΩi are both connected, assumption (18) and Theorem 2.3 (i) imply that there exists a unique η ∈ C m−1,α (∂Ωi , Rn ) such that 1 η(t) + v∗ [ω, η](t) = −g i (t) ∀t ∈ ∂Ωi . (27) 2 R R By integrating (27), we obtain ∂Ωi η dσ = − ∂Ωi g i dσ (cf. e.g., [3, right above Thm. 6.6].) Now let Z ] − u (x) ≡ v [ω, η](x) + δ2,n Γn (ω, x) g i (t) dσt ∂Ωi Z (Γn (ω, x − t) − δ2,n Γn (ω, x))η(t) dσt ∀x ∈ Rn \ Ωi . = ∂Ωi

If n ≥ 3, then u] satisfies the inequalities in the last two conditions of problem (26) by the definition of Γn . If n = 2, the boundedness properties of the functions in (13) imply that u] satisfies the last two conditions of problem (26). Since both ui and v − [ω, η] satisfy the equation −T (ω, Du)ν i = g i

on ∂Ωi ,

Theorem 2.3 (ii) implies that ui = v − [ω, η]. Hence, equation (27) and standard jump properties of elastic layer potentials imply that η satisfies equation M2 [0, 0, 0, γm , c, η, ρ] = 0. Then Theorem 2.3 (i) and equation (27) guarantee that η is uniquely determined. If γm = 0, then Theorem 2.3 (i) implies that η = 0, and thus ui0 [η] = 0. Hence, the statement follows. 2 We shall now show that if the limiting problem (26) has a solution u ˜i satisfying a certain nondegeneracy condition, then for sufficiently small, problem (2) has a solution which is unique in a sense which we clarify below. Theorem 3.3. Let α ∈]0, 1[, ω ∈]1 − (2/n), +∞[, m ∈ N \ {0}. Let Ωi , Ωo be as in (15). Let 0 be as in (16). Let Gi , Go be as in (17), (18), (19), (20). Let γ(·) be a map of ]0, 0 [ to ]0, +∞[. Let (3) hold. Let FGi be real analytic in C m−1,α (∂Ωi , Rn ) .

(28)

Assume that −ao satisfies conditions (6), (7) on ∂Ωo . Let (˜ c, ρ˜) be the only pair in Rn × C m−1,α (∂Ωo , Rn )0 such that v + [ω, ρ˜] + c˜ is the only solution u ˜ in C m,α (clΩo , Rn ) of (1). Let u ˜i be a solution of the limiting problem (26) in m,α Cloc (Rn \Ωi , Rn ). Let η˜ be the only element of C m−1,α (∂Ωi , Rn ) which satisfies 14

(25), and such that ui0 [˜ η] = u ˜i . Let G i be the matrix valued function on ∂Ωi defined by i i G (t) ≡ −Dξ G t, (1 − δ2,n )˜ ui (t) Z δ2,n ω + 2 i i T (ω, D˜ u )ν dσ + u ˜(0) ∀t ∈ ∂Ωi . + 4π ω + 1 ∂Ωi R m (ω+2) If n = 2, we assume that the matrix I − γ4π(ω+1) G i dσ is invertible. ∂Ωi If n ≥ 3 and γm > 0, we assume that G i satisfies assumptions (6), (7) on i ∂Ω (while in case γm = 0, we do not assume that G i satisfies (6), (7)). Then there exist 0 ∈]0, 0 [ and an open neighborhood Uγm of (0, 0, γm ) in 3 R , and an open neighborhood V of (˜ c, η˜, ρ˜) in Rn × Xm,α and a real analytic 0 0 operator (C, E, R) of ] − , [×Uγm to V such that (log )δ2,n n−2 , ∈ Uγm ∀ ∈]0, 0 [ (29) (log )δ2,n , (log )δ2,n γ() holds and the set of zeros of M in ] − 0 , 0 [×Uγm × V coincides with the graph of (C, E, R). In particular, (C[0, 0, 0, γm ], E[0, 0, 0, γm ], R[0, 0, 0, γm ]) = (˜ c, η˜, ρ˜) . Proof. We plan to apply the Implicit Function Theorem to equation M = 0 around the point (0, 0, 0, γm , c˜, η˜, ρ˜). By assumption (28), and by standard properties of elastic layer potentials (cf. e.g., [3, Appendix]), and by known properties of (nonsingular) integral operators (cf. e.g., [12, Thm. 6.2]), we conclude that the map M is real analytic. By definition of (˜ c, η˜, ρ˜), we have M [0, 0, 0, γm , c˜, η˜, ρ˜] = 0. Then by standard calculus in Banach space (see also [12, Prop. 6.3]), the differential of M at the point (0, 0, 0, γm , c˜, η˜, ρ˜) with respect to the variable (c, η, ρ) is delivered by the formula Z ∂(c,η,ρ) M1 [0, 0, 0, γm , c˜, η˜, ρ˜](c, η, ρ) = ρ dσ , (30) ∂Ωo

1 ∂(c,η,ρ) M2 [0, 0, 0, γm , c˜, η˜, ρ˜](c, η, ρ) = η + v∗ [ω, η] 2 Z δ2,n ω + 2 η dσ + v[ω, ρ](0) + c −γm G i · (1 − δ2,n )v[ω, η] + 4π ω + 1 ∂Ωi on ∂Ωi , ∂(c,η,ρ) M3 [0, 0, 0, γm , c˜, η˜, ρ˜](c, η, ρ) 1 = − ρ + v∗ [ω, ρ] − ao · {v[ω, ρ] + c} 2

on ∂Ωo ,

for all (c, η, ρ) ∈ Rn × Xm,α . We now prove that ∂(c,η,ρ) M [0, 0, 0, γm , c˜, η˜, ρ˜] is a linear homeomorphism of Rn × Xm,α onto itself. By the Open Mapping Theorem, it suffices to show that ∂(c,η,ρ) M [0, 0, 0, γm , c˜, η˜, ρ˜] is a bijection of 15

Rn × Xm,α onto itself. Let (d, f i , f o ) ∈ Rn × Xm,α . We must show that there exists a unique (c, η, ρ) in Rn × Xm,α such that ∂(c,η,ρ) M [0, 0, 0, γm , c˜, η˜, ρ˜](c, η, ρ) = (d, f i , f o ) .

(31)

By Theorem 2.2 (i), the first and third component of (31) admit a unique solution (c, ρ) ∈ Rn × C m−1,α (∂Ωo , Rn ). We now consider separately cases n = 2 and n ≥ 3, and we start with case n = 2. By Theorem 2.3 (i), the map which takes η to 21 η + v∗ [ω, η] is a linear homeomorphism of C m−1,α (∂Ωi , R2 ). By assumption (28), and by the known form of the differential of a composition operator, G i must be an element of C m−1,α (∂Ωi , M2 (R)) (see also [12, Prop. 6.3].) Thus the map Ψ[·] of C m−1,α (∂Ωi , R2 ) to itself which takes η to Z 1 1 ω+2 Ψ[η] ≡ η + v∗ [ω, η] − γm G i · η dσ , (32) 2 4π ω + 1 ∂Ωi is a compact perturbation of an isomorphism, which is a Fredholm operator of index 0, and thus Ψ is Fredholm of index 0 (cf. e.g., Deimling [6, Thm. 9.8, p. 79].) We now show that Ker Ψ = {0}, a condition which implies that Ψ is an isomorphism. Let η ∈ C m−1,α (∂Ωi , R2 ) be such that Ψ[η] = 0. Then by integrating the function in the right hand side of (32) on ∂Ωi and by exploiting the equality Z Z 1 η + v∗ [ω, η] dσ = η dσ , 2 ∂Ωi ∂Ωi (cf. e.g., [3, right above Thm. 6.6]), we obtain Z Z 1 ω+2 I − γm G i dσ · η dσ = 0 . 4π ω + 1 ∂Ωi ∂Ωi R Since the matrix in parentheses is invertible, we must have ∂Ωi η dσ = 0. Then by equality Ψ[η] = 0, we deduce that 21 η + v∗ [ω, η] = 0, which in turn implies η = 0 (see Theorem 2.3 (i).) Then Ψ is injective and thus Ψ is an isomorphism of C m−1,α (∂Ωi , R2 ) onto itself. Thus it follows that the second component of (31) admits a unique solution η ∈ C m−1,α (∂Ωi , R2 ). We now consider case n ≥ 3. We must show that equation 1 η + v∗ [ω, η] − γm G i · v[ω, η] = f i + γm G i · {v[ω, ρ](0) + c} , 2 has a unique solution η ∈ C m−1,α (∂Ωi , Rn ). If γm > 0, we have assumed that G i satisfies conditions (6), (7). Then Theorem 2.2 (v) implies that η exists and is unique. If γm = 0, the same conclusion holds by Theorem 2.3 (i). Then we can apply the Implicit Function Theorem for real analytic operators to equation M = 0 and deduce the existence of 0 , Uγm , V and (C, E, R) (cf. e.g., Deimling [6, Thm. 15.3].) Possibly shrinking 0 , we can clearly assume that (29) holds. 2 We note that if n ≥ 3, and if γ −1 () has a real analytic extension to a whole neighborhood of 0, then possibly shrinking 0 , the map 7→ (C[, , n−2 , γ −1 ()], E[, , n−2 , γ −1 ()], R[, , n−2 , γ −1 ()]) 16

has a real analytic extension on ] − 0 , 0 [. In order to simplify the notation of (29), we introduce the function Ξm,n of ]0, 0 [ to Uγm by setting (log )δ2,n n−2 , Ξm,n [] ≡ (log )δ2,n , ∀ ∈]0, 0 [ . (33) (log )δ2,n γ() We are now ready to define our family of solutions. Definition 3.4. Let the assumptions of Theorem 3.3 hold. Then we set u(, t) ≡ u [C[, Ξm,n []], E[, Ξm,n []], R[, Ξm,n []]] (t)

∀t ∈ cl Ω() ,

for all ∈]0, 0 [. (cf. Theorem 3.1.) In this paper, we do not investigate the best conditions of solvability of the limiting problem (26). We just observe that if γm Gi (t, ξ) is a small perturbation of a function of the form −B(t)ξ with B ∈ C m−1,α (∂Ωi , Mn (R)) such that Z 1 ω+2 B dσ 6= 0 if n = 2 , det I − 4π ω + 1 ∂Ωi ξ t B(t)ξ > 0

∀t ∈ ∂Ωi , ∀ξ ∈ Rn \ {0}

if n ≥ 3 ,

one could exploit a standard argument based on the Leray-Schauder topological degree to show that under suitable regularity assumptions on Gi equation (25) and thus the limiting problem (26) does actually have solutions as in the statement of Theorem 3.3.

4

A functional analytic representation Theorem for the family {u(, ·)}∈]0,0 [ and for its energy integral.

Theorem 4.1. Let the assumptions of Theorem 3.3 hold. Then the following statements hold. ˜ be a bounded open subset of Ωo \ {0} such that 0 ∈ ˜ Then (i) Let Ω / clΩ. there exist Ω˜ ∈]0, 0 [ and a real analytic operator UΩ˜ of ] − Ω˜ , Ω˜ [×Uγm ˜ Rn ) such that Ω ˜ ⊆ Ω() for all ∈] − ˜ , ˜ [ and such that to C m,α (clΩ, Ω Ω u(, t)|clΩ˜ = UΩ˜ [, Ξm,n []] (t)

˜, ∀t ∈ clΩ

(34)

for all ∈]0, Ω˜ [ (see also (33).) Moreover, UΩ˜ [0, 0, 0, γm ] = u ˜|clΩ˜ and thus lim u(, ·)|clΩ˜ = u ˜|clΩ˜ (·)

→0+

(35)

˜ Rn ), where u in C m,α (clΩ, ˜ is the unique solution in C m,α (clΩo , Rn ) of (1). 17

˜ be a bounded open subset of Rn \ clΩi . Then there exist ˜ ∈]0, 0 [, (ii) Let Ω Ω,1 m,α ˜ Rn ) such (clΩ, and a real analytic map UΩ˜r of ] − Ω,1 ˜ , Ω,1 ˜ [×Uγm to C that ˜ ⊆ 1 Ω() Ω ∀ ∈] − Ω,1 ˜ , Ω,1 ˜ [\{0} and such that ˜, ∀t ∈ clΩ

u(, t) = UΩ˜r [, Ξm,n []] (t)

(36)

for all ∈]0, Ω,1 ˜ [. Moreover, UΩ˜r [0, 0, 0, γm ]

= (1 −

δ2,n )˜ ui|clΩ˜

δ2,n ω + 2 + 4π ω + 1

Z

T (ω, D˜ ui )ν i dσ + u ˜(0) ,

∂Ωi

(37) ˜ Rn ), where u and lim→0+ u(, ·) = UΩ˜r [0, 0, 0, γm ] in C m,α (clΩ, ˜i is as in Theorem 3.3. ˜ ⊆ Ω() for Proof. We first prove statement (i). Let ∗ ∈]0, 0 [ be such that Ω ˜ Ω

all ∈ [−∗Ω˜ , ∗Ω˜ ]. Let Ω˜ ∈]0, ∗Ω˜ [ be such that clΩi ⊆ ∗Ω˜ Ωi for all ∈ [−Ω˜ , Ω˜ ]. By definition of u(, ·), we have Z n−2 u(, t) = Γn (ω, t − s)E[, Ξm,n []](s) dσ (log )δ2,n ∂Ωi +v[ω, R[, Ξm,n []](t) + C[, Ξm,n []] ∀t ∈ clΩ() , for all ∈]0, Ω˜ [. Thus it is natural to define Z UΩ(∗˜ ) [, 1 , 2 , 3 ](t) ≡ 2 Γn (ω, t − s)E[, 1 , 2 , 3 ](s) dσs Ω

(38)

∂Ωi

+v[ω, R[, 1 , 2 , 3 ]](t) + C[, 1 , 2 , 3 ] ∀t ∈ clΩ(∗Ω˜ ) for all (, 1 , 2 , 3 ) ∈] − Ω˜ , Ω˜ [×Uγm . Thus we are reduced to show that the right hand side of (38) defines a real analytic operator of ] − Ω˜ , Ω˜ [×Uγm ˜ ⊆ Ω(∗ ) and thus we can take U ˜ equal to C m,α (clΩ(∗Ω˜ ), Rn ). Indeed, Ω ˜ Ω Ω ˜ Since to UΩ(∗˜ ) composed with the restriction of operator of clΩ(∗Ω˜ ) to clΩ. Ω Ω(∗Ω˜ ) ⊆ clΩo , known regularity properties of the elastic potentials (cf. e.g., [3, Thm. 6.2]), and the real analyticity of R implies that the map of ]−Ω˜ , Ω˜ [×Uγm to C m,α (clΩo , Rn ), which takes (, 1 , 2 , 3 ) to v + [ω, R[, 1 , 2 , 3 ]] is real analytic. By standard properties of integral operators with real analytic kernel and 1 i n with no singularity (see also [12, Prop. 6.1]), the map of R ] − Ω˜ , Ω˜ [×L (∂Ω , R ) m+1 ∗ n to C (clΩ(Ω˜ ), R ) which takes (, f ) to the map ∂Ωi Γn (ω, t − s)f (s) dσs of the variable t ∈ clΩ(∗Ω˜ ) is real analytic. Since E is real analytic from ] − Ω˜ , Ω˜ [×Uγm to C m−1,α (∂Ωi , Rn ) and since C m−1,α (∂Ωi , Rn ) is continuously imbedded into L1 (∂Ωi , Rn ) and C m+1 (clΩ(∗Ω˜ ), Rn ) is continuously imbedded into C m,α (clΩ(∗Ω˜ ), Rn ), we conclude that the function of ] − Ω˜ , Ω˜ [×Uγm to C m,α (clΩ(∗Ω˜ ), Rn ) which takes (, 1 , 2 , 3 ) to the function Z Γn (ω, t − s)E[, 1 , 2 , 3 ](s) dσs ∂Ωi

18

of the variable t ∈ clΩ(∗Ω˜ ) is real analytic. Also, Theorems 3.2 and 3.3 and definition (38) imply that c=u ˜ on clΩ(∗Ω˜ ) . UΩ(∗˜ ) [0, 0, 0, γm ] = v[ω, R[0, 0, 0, γm ]]+C[0, 0, 0, γm ] = v[ω, ρ˜]+˜ Ω

˜ ⊆ 1 Ωo for all ∈ We now prove statement (ii). Let ∗Ω,1 ∈]0, 0 [ be such that Ω ˜ 1 m,α n ∗ ∗ ∗ ] \ {0}. Since the restriction operator from C ), R , clΩ( [−Ω,1 ˜ ˜ ˜ ∗ Ω,1 Ω,1 ˜ Ω,1

˜ Rn ) is linear and continuous, it clearly suffices to construct first to C m,α (clΩ, r U ∗1 Ω(∗ ) and then to define UΩ˜r to be the composition of U r∗1 Ω(∗ ) with the ˜ Ω,1

˜ Ω,1

restriction operator of

∗ ˜ ) ∗˜ clΩ(Ω,1 Ω,1 1

˜ > 0 be such that interval. Let R

˜ Ω,1

˜ Ω,1

˜ for ranging in a possibly smaller to clΩ

1 ∗˜

Ω,1

˜ Ωo ⊆ Bn (0, R/2). Let Ω,1 ∈]0, ∗Ω,1 ˜ ˜ ] be

˜ ˜ ) ⊆ Ωo , so that such that clBn (0, R Ω,1 1 o ˜ ˜ ⊆ clBn (0, R) ˜ ⊆ 1 Ωo ⊆ Bn (0, R) Ω ⊆ Bn (0, R/2) ∗Ω,1 ˜ for all ∈] − Ω,1 ˜ , Ω,1 ˜ [\{0}. By definition of u(, ·), we have Z 1 δ2,n ω + 2 u(, t) = v[ω, E[, Ξm,n []]](t) + E[, Ξm,n []] dσ (log )δ2,n 4π ω + 1 ∂Ωi +v[ω, R[, Ξm,n []]](t) + C[, Ξm,n []] 1 ∗˜

for all t ∈

Ω,1

U r∗1

˜ Ω,1

clΩ(∗Ω,1 ˜ [. Thus it is natural to set ˜ ) and ∈]0, Ω,1

Ω(∗˜ ) [, 1 , 2 , 3 ](t)

≡ (1 − δ2,n (2 − 1))v[ω, E[, 1 , 2 , 3 ]](t)

Ω,1

Z δ2,n ω + 2 E[, 1 , 2 , 3 ] dσ 4π ω + 1 ∂Ωi +v[ω, R[, 1 , 2 , 3 ]](t) + C[, 1 , 2 , 3 ] , +

for all t ∈

1 ∗˜

Ω,1

clΩ(∗Ω,1 ˜ , Ω,1 ˜ [×Uγm . By standard ˜ ) and all (, 1 , 2 , 3 ) ∈] − Ω,1

properties of elastic single layer potentials and by Theorem 3.3, the map which takes (, 1 , 2 , 3 ) in ] − Ω,1 ˜ , Ω,1 ˜ [×Uγm to v[ω, R[, 1 , 2 , 3 ]] in the space 0 CL[ω] (clΩo , Rn ) ≡ {u ∈ C 0 (clΩo , Rn ) ∩ C 2 (Ωo , Rn ) : L[ω]u = 0}

endowed with the sup-norm is real analytic. Then by Proposition 6.2 of the 0 n ˜ Appendix, the map of ] − Ω,1 ˜ , Ω,1 ˜ [×Uγm to the space CL[ω] (clBn (0, R), R ) + which takes (, 1 , 2 , 3 ) to the map v [ω, R[, 1 , 2 , 3 ]](t) of the variable ˜ is real analytic. By classical interior estimates for the solutions of t ∈ clBn (0, R) 0 ˜ Rn ) equation L[ω]u = 0, one can easily see that the restriction of CL[ω] (clBn (0, R), n to C m,α ( ∗1 clΩ(∗Ω,1 ˜ ), R ) is real analytic (see Theorem 6.1 of the Appendix.) ˜ Ω,1

19

Then the map which takes (, 1 , 2 , 3 ) to the function v + [ω, R[, 1 , 2 , 3 ]](t) of the variable t ∈ ∗1 clΩ(∗Ω,1 ˜ , Ω,1 ˜ [×Uγm to ˜ ) is real analytic from ] − Ω,1 ˜ Ω,1

n C m,α ( ∗1 clΩ(∗Ω,1 ˜ ), R ). Then by the real analyticity of E, C, we deduce that

U r∗1

˜ Ω,1

˜ Ω,1

Ω(∗˜ )

is real analytic. By Theorems 3.2 and 3.3 and by equality

Ω,1

Z

i

i

Z

T (ω, D˜ u )ν dσ = ∂Ωi

∂Ωi

1 η˜ + v∗ [ω, η˜] 2

Z dσ =

η˜ dσ , ∂Ωi

which follows by standard properties of elastic layer potentials (cf. e.g., [3, right above Thm. 6.6]), we have u ˜ = v[ω, R[0, 0, 0, γm ]] + C[0, 0, 0, γm ] and U r∗1

˜ Ω,1

Ω(∗˜ ) [0, 0, 0, γm ]

= (1 − δ2,n )v[ω, E[0, 0, 0, γm ]]

Ω,1

Z δ2,n ω + 2 E[0, 0, 0, γm ] dσ 4π ω + 1 ∂Ωi +v[ω, R[0, 0, 0, γm ]](0) + C[0, 0, 0, γm ] Z δ2,n ω + 2 i = (1 − δ2,n )˜ u| ∗1 clΩ(∗ ) + T (ω, D˜ ui )ν i dσ + u ˜(0) . ˜ 4π ω + 1 Ω,1 i ˜ ∂Ω Ω,1 +

2

Thus the proof is complete.

We now consider the energy integral of the family {u(, ·)}∈]0,0 [ , and we prove the following. Theorem 4.2. Let the assumptions of Theorem 3.3 hold. Then there exist ˜ ∈]0, 0 [ and a real analytic map F of ] − ˜, ˜[×Uγm to R such that Z 1 t tr T (ω, Dx u(, x))(Dx u(, x)) dx (39) F [, Ξm,n []] = 2 Ω() for all ∈]0, ˜[ (see also (33).) Moreover, Z 1 F[0, 0, 0, γm ] = tr T (ω, D˜ u)(D˜ u)t dx , 2 Ωo where u ˜ is the unique solution in C m,α (clΩo , Rn ) of (1).

20

Proof. By the Divergence Theorem, we have Z t tr T (ω, Dx u(, x))(Dx u(, x)) dx (40) Ω() Z =− ut (, s)T (ω, Ds u(, s))νΩi (s) dσs ∂Ωi Z + ut (, s)T (ω, Ds u(, s))ν o (s) dσs o ∂Ω Z 1 ut (, s)Gi (s/, u(, s)) dσs = γ() ∂Ωi Z + ut (, s){ao (s) · u(, s) + g(s)} dσs ∂Ωo Z n−1 = ut (, s)Gi (s, u(, s)) dσs γ() ∂Ωi Z + ut (, s){ao (s) · u(, s) + g(s)} dσs ∂Ωo

o n for all ∈]0, 0 [. Hence, it suffices to take ˜ ≡ min Ω(0 ) , 10 Ω(0 ),1 and to set

F[, 1 , 2 , 3 ] Z t 1 U r10 Ω(0 ) [, 1 , 2 , 3 ](s) Gi s, U r10 Ω(0 ) [, 1 , 2 , 3 ](s) dσs ≡ 2 3 2 i Z ∂Ω t 1 + UΩ(0 ) [, 1 , 2 , 3 ](s) a(s) · UΩ(0 ) [, 1 , 2 , 3 ](s) + g(s) dσs 2 ∂Ωo for all (, 1 , 2 , 3 ) ∈] − ˜, ˜[×Uγm (see Theorem 4.1.) Finally, equations (34), (36), (40) imply that formula (39) holds. The formula for F[0, 0, 0, γm ] follows by the definition of F and by Theorem 4.1 (see (35).) 2

5

A property of local uniqueness for the family {u(, ·)}∈]0,0 [

We now show by means of the following theorem, that the family {u(, ·)}∈]0,0 [ is essentially unique. Theorem 5.1. Let the assumptions of Theorem 3.3 hold. If {εj }j∈N is a sequence of ]0, 0 [ converging to 0 and if {uj }j∈N is a sequence of functions such that uj ∈ C m,α (clΩ(εj ), Rn ) , uj solves (2) for = εj , limj→∞ uj (εj ·)|∂Ωi = (1 − δ2,n )˜ ui (·) + m−1,α i n in C (∂Ω , R ) , 21

δ2,n ω+2 4π ω+1

R ∂Ωi

T (ω, D˜ ui )ν i dσ + u ˜(0)

then there exists j0 ∈ N such that uj (·) = u(εj , ·) for all j ≥ j0 . Proof. Since uj solves (2), Theorems 3.1, 3.2 ensure that there exist (cj , ηj , ρj ) and (˜ c, η˜, ρ˜) in Rn × Xm,α such that M [εj , Ξm,n [εj ], cj , ηj , ρj ] = 0 ,

M [0, 0, 0, γm , c˜, η˜, ρ˜] = 0 ,

and that uj = v + [ω, µj ] + cj ,

u ˜ = v + [ω, ρ˜] + c˜ ,

u ˜i = v − [ω, η˜],

where µj (y) = ρj (y)

if y ∈ ∂Ωo ,

µj (y) =

1 ηj (y/εj ) εj (log εj )δ2,n

if y ∈ εj ∂Ωi .

We now rewrite equation M [, 1 , 2 , 3 , c, η, ρ] = 0 in the following form M1 [, 1 , 2 , 3 , c, η, ρ] = 0 , (41) Z n X 1 ρl (s)T (ω, Dξ Γln (ω, t − s))ν i (t) dσs η(t) + v∗ [ω, η](t) + 1 2 o ∂Ω l=1 −3 G i (t) · (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 + η dσ + v[ω, ρ](t) + c 4π ω + 1 ∂Ωi = −3 Gi t, (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 + η dσ + v[ω, ρ](t) + c 4π ω + 1 ∂Ωi −3 G i (t) · (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 + η dσ + v[ω, ρ](t) + c ∀t ∈ ∂Ωi , 4π ω + 1 ∂Ωi Z n X 1 ηl (s)T (ω, Dξ Γln (ω, t − s))ν o (t) dσs − ρ(t) + v∗ [ω, ρ](t) + 2 2 ∂Ωi l=1 Z −ao (t) 2 Γn (ω, t − s)η(s) dσs + v[ω, ρ](t) + c = g(t) ∀t ∈ ∂Ωo . ∂Ωi

Next we denote by N [, 1 , 2 , 3 , c, η, ρ] ≡ (Nl [, 1 , 2 , 3 , c, η, ρ])l=1,2,3 the function of (, 1 , 2 , 3 , c, η, ρ) ∈] − 0 , 0 [×Rn+3 × Xm,α to Rn × Xm,α defined by N1 ≡ M1 and such that N2 and N3 equal the left hand side of the second and

22

the third equation in (41), respectively. Thus equation (41) can be rewritten as N1 [, 1 , 2 , 3 , c, η, ρ] = 0

(42)

N2 [, 1 , 2 , 3 , c, η, ρ](t) = −3 Gi t, (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 + η dσ + v[ω, ρ](t) + c 4π ω + 1 ∂Ωi −3 G i (t) · (1 + δ2,n (2 − 1))v[ω, η](t) Z δ2,n ω + 2 + η dσ + v[ω, ρ](t) + c ∀t ∈ ∂Ωi , 4π ω + 1 ∂Ωi N3 [, 1 , 2 , 3 , c, η, ρ] = g on ∂Ωo . By our assumption on FGi , and by the known form of the differential of a composition operator, G i must be an element of C m−1,α (∂Ωi , Mn (R)) (see [12, Prop. 6.3], where the scalar case has been worked out, but the proof is the same for matrix valued functions.) Then by standard properties of integral operators with real analytic kernel and with no singularity (see [12, Thm. 6.2]), and by standard properties of elastic layer potentials (cf. e.g., [3, Thm. 6.2]), the map N is real analytic. Next we note that N [, 1 , 2 , 3 , ·, ·, ·] is linear for all fixed (, 1 , 2 , 3 ) ∈] − 0 , 0 [×R3 . Accordingly, the map of ] − 0 , 0 [×R3 to L(Rn × Xm,α , Rn × Xm,α ) which takes (, 1 , 2 , 3 ) to N [, 1 , 2 , 3 , ·, ·, ·] is real analytic. We also note that N [0, 0, 0, γm , ·, ·, ·] = ∂(c,η,ρ) M [0, 0, 0, γm , c˜, η˜, ρ˜](·, ·, ·) , and thus that N [0, 0, 0, γm , ·, ·, ·] is a linear homeomorphism (see the proof of Theorem 3.3.) Since the set of linear homeomorphisms is open in the set of linear and continuous operators, and since the map which takes a linear invertible operator to its inverse is real analytic (cf. e.g., Hille and Phillips [9, Thms. 4.3.2 and 4.3.4]), there exists an open neighborhood W of (0, 0, 0, γm ) in ] − 0 , 0 [×R3 such that the map (, 1 , 2 , 3 ) 7→ N [, 1 , 2 , 3 , ·, ·, ·](−1) is real analytic from W to L(Rn × Xm,α , Rn × Xm,α ). Clearly, there exists j1 ∈ N such that (εj , Ξm,n [εj ]) ∈ W for all j ≥ j1 . Since M [εj , Ξm,n [εj ], cj , ηj , ρj ] = 0, the invertibility of N [εj , Ξm,n [εj ], ·, ·, ·] and equality (42) guarantee that (−1) (cj , ηj , ρj ) = N [εj , Ξm,n [εj ], ·, ·, ·] 0, εj (log εj )δ2,n i i i i − FG [uj (εj ·)|∂Ω ] + G · uj (εj ·)|∂Ω , g γ(εj )

23

for all j ≥ j1 . By assumption (28), FGi [·] is continuous in C m−1,α (∂Ωi , Rn ). By assumption (3), lim→0 γ −1 ()(log )δ2,n = γm ∈ [0, +∞[. Hence, εj (log εj )δ2,n εj (log εj )δ2,n i FGi [uj (εj ·)|∂Ωi ] − G · uj (εj ·)|∂Ωi (43) j→∞ γ(εj ) γ(εj ) Z δ2,n ω + 2 = −γm FGi (1 − δ2,n )˜ ui (·) + T (ω, D˜ ui )ν i dσ + u ˜(0) 4π ω + 1 ∂Ωi Z δ2,n ω + 2 i i i i −γm G · (1 − δ2,n )˜ u (·) + T (ω, D˜ u )ν dσ + u ˜(0) 4π ω + 1 ∂Ωi lim −

in C m−1,α (∂Ωi , Rn ). The analyticity of (, 1 , 2 , 3 ) 7→ N [, 1 , 2 , 3 , ·, ·, ·](−1) guarantees that lim N [εj , Ξm,n [εj ], ·, ·, ·]

(−1)

j→∞

= N [0, 0, 0, γm , ·, ·, ·](−1) ,

(44)

in L(Rn × Xm,α , Rn × Xm,α ). Since the evaluation map of L(Rn × Xm,α , Rn × Xm,α ) × (Rn × Xm,α ) to Rn × Xm,α , which takes a pair (A, v) to A[v] is bilinear and continuous, the limiting relations of (43) and (44) imply that (−1) lim (cj , ηj , ρj ) = lim N [εj , Ξm,n [εj ], ·, ·, ·] 0, (45) j→∞ j→∞ εj (log εj )δ2,n FGi [uj (εj ·)|∂Ωi ] + G i · uj (εj ·)|∂Ωi , g − γ(εj ) = N [0, 0, 0, γm , ·, ·, ·](−1) Z δ2,n ω + 2 0, −γm FGi (1 − δ2,n )˜ ui (·) + T (ω, D˜ ui )ν i dσ + u ˜(0) 4π ω + 1 ∂Ωi Z δ 2,n ω + 2 −γm G i · (1 − δ2,n )˜ ui (·) + T (ω, D˜ ui )ν i dσ + u ˜(0) , g 4π ω + 1 ∂Ωi

in Rn × Xm,α . Since M [0, 0, 0, γm , c˜, η˜, ρ˜] = 0, the right hand side of (45) equals (˜ c, η˜, ρ˜). Hence, lim (εj , Ξm,n [εj ], cj , ηj , ρj ) = (0, 0, 0, γm , c˜, η˜, ρ˜)

j→∞

in Rn+4 × Xm,α . Thus Theorem 3.3 implies that there exists j0 ∈ N such that cj = C [εj , Ξm,n [εj ]] ,

ηj = E [εj , Ξm,n [εj ]] ,

ρj = R [εj , Ξm,n [εj ]] ,

for all j ≥ j0 . Accordingly, uj (·) = u(εj , ·) for j ≥ j0 (see Definition 3.4).

6

2

Appendix

We first introduce the Romieu classes. For all bounded open subsets Ω of Rn and ρ > 0, we set ρ|β| Cρ0 (clΩ) ≡ u ∈ C ∞ (clΩ) : sup kDβ ukC 0 (clΩ) < +∞ , β∈Nn |β|! 24

and

ρ|β| kDβ ukC 0 (clΩ) ∀u ∈ Cρ0 (clΩ) . β∈Nn |β|! As is well known, the Romieu class Cρ0 (clΩ), k · kCρ0 (clΩ) is a Banach space. Then we note that we have the following immediate consequence of the wellknown a-priori estimates for elliptic systems of Douglis and Nirenberg [7, Thm. 1] and of the Stirling inequality. kukCρ0 (clΩ) ≡ sup

Theorem 6.1. Let ω ∈]1 − (2/n), +∞[. Let m ∈ N. Let Ω, Ω1 be bounded open subsets of Rn with clΩ1 ⊆ Ω. Let d be the distance between clΩ1 and Rn \ Ω. Let 0 (clΩ, Rn ) ≡ u ∈ C 0 (clΩ, Rn ) ∩ C 2 (Ω, Rn ) : L[ω]u = 0 in Ω . CL[ω] Then there exists c > 0 such that |β| c|β| |Dβ u(x)| ≤ sup |u(y)| , r y∈Bn (x,r)

(46)

0 for all x ∈ clΩ1 , r ∈]0, d[ and for all β ∈ Nn and for all u ∈ CL[ω] (clΩ, Rn ). In particular, there exists ρ > 0 such that the restriction operator of the space n 0 CL[ω] (clΩ, Rn ) endowed with the norm of the uniform convergence to Cρ0 (clΩ1 ) and thus to C m+1 (clΩ1 , Rn ) is linear and continuous.

Proof. By classical elliptic interior estimates applied to the ball Bn (0, 1) there exists c > 0 such that |Dxj u(0)| ≤ c supy∈Bn (0,1) |u(y)| for all solutions u of class C 2 of equation L[ω]u = 0 defined in an open neighborhood of clBn (0, 1) and for all j = 1, . . . , n (cf. Douglis and Nirenberg [7, Thm. 1].) Then we immediately deduce that |Dxj u(x)| ≤ rc supy∈Bn (x,r) |u(y)| for all x ∈ clΩ1 , 0 r ∈]0, d[ and for all u ∈ CL[ω] (clΩ, Rn ). Then by a standard inductive argument, we deduce that (46) holds. Then by the Stirling formula, there exists ρ > 0 such that the restriction operator of the statement is linear and continuous. 2 Then we deduce the validity of the following statement on the composition operator by Theorem 6.1 and by Proposition 9 of [13] (we note that Proposition 9 of [13] is in turn a variant of a result of Preciso [18, Prop. 1.1, p. 101].) Proposition 6.2. Let n1 , n2 ∈ N\{0}. Let Ω1 be a bounded open subset of Rn1 . Let Ω2 be a bounded open connected subset of Rn2 . Let ω ∈]1 − (2/n1 ), +∞[. 0 Let the space CL[ω] (clΩ1 , Rn1 ) be endowed with the norm of the uniform conver0 gence. Then the composition operator F of CL[ω] (clΩ1 , Rn1 ) × C 0 (clΩ2 , Ω1 ) to 0 n1 C (clΩ2 , R ) defined by F [u, v] ≡ u ◦ v

0 ∀(u, v) ∈ CL[ω] (clΩ1 , Rn1 ) × C 0 (clΩ2 , Ω1 ) ,

is real analytic. We note that Proposition 9 of [13] has been proved for n1 = n2 . However, case n1 6= n2 holds with the same proof. 25

Acknowledgement We acknowledge the support of the research project “Problemi di stabilit`a per operatori differenziali” of the University of Padova, Italy.

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27