On the Asymptotic Behavior of D-Solutions to the Displacement Problem of Linear Elastostatics in Exterior Domains

The displacement problem of elastostatics in an exterior Lipschitz domain Ω of R3 consists of finding a solution to the equations [1] (Notation—Unless otherwise specified, we will use the notation of the classical monograph [1] by M.E. Gurtin. In particular, (div C[∇u])i = ∂j(Cijhk∂kuh), Lin is the space of second–order tensors (linear maps from R3 into itself) and Sym, Skw are the spaces of the symmetric and skew elements of Lin respectively; if E ∈ Lin and v ∈ R3, Ev is the vector with components Eijvj. BR = {x ∈ R3 : r = |x| < R}, TR = B2R \ BR, {BR = R3 \ BR and BR0 is a fixed ball containing ∂Ω. If f (x) and φ(r) are functions defined in a neighborhood of infinity, then f (x) = o(φ(r)) means that limr→+∞( f /φ) = 0. To lighten up the notation, we do not distinguish between scalar, vector, and second–order tensor space functions; c will denote a positive constant whose numerical value is not essential to our purposes.)


Introduction
The displacement problem of elastostatics in an exterior Lipschitz domain Ω of R 3 consists of finding a solution to the equations [1] (Notation-Unless otherwise specified, we will use the notation of the classical monograph [1] by M.E. Gurtin. In particular, (div C[∇u]) i = ∂ j (C ijhk ∂ k u h ), Lin is the space of second-order tensors (linear maps from R 3 into itself) and Sym, Skw are the spaces of the symmetric and skew elements of Lin respectively; if E ∈ Lin and v ∈ R 3 , Ev is the vector with components E ij v j . B R = {x ∈ R 3 : r = |x| < R}, T R = B 2R \ B R , B R = R 3 \ B R and B R 0 is a fixed ball containing ∂Ω. If f (x) and φ(r) are functions defined in a neighborhood of infinity, then f (x) = o(φ(r)) means that lim r→+∞ ( f /φ) = 0. To lighten up the notation, we do not distinguish between scalar, vector, and second-order tensor space functions; c will denote a positive constant whose numerical value is not essential to our purposes.) div C[∇u] = 0 in Ω, where u is the (unknown) displacement field,û is an (assigned) boundary displacement, B is the unit ball, C ≡ [C ijhk ] is the (assigned) elasticity tensor, i.e., a map from Ω × Lin → Sym, linear on Sym and vanishing in Ω × Skw. We shall assume C to be symmetric, i.e., and positive definite, i.e., there exists positive scalars µ 0 and µ e (minimum and maximum elastic moduli [1]) such that Let D 1,q (Ω), D 1,q 0 (Ω) (q ∈ [1, +∞)) be the completion of C ∞ 0 (Ω) and C ∞ 0 (Ω), respectively, with respect to the norm ∇u L q (Ω) .
We consider solutions u to equations (1) with finite Dirichlet integral (or with finite energy) that we call D-solutions analogous with the terminology used in viscous fluid dynamics (see [2]). More precisely, we say that u ∈ D 1,2 (Ω) is a D-solution to equation (1) 1 A D-solution to system (1) is a D-solution to equation (1) 1 , which satisfies the boundary condition in the sense of trace in Sobolev's spaces and tends to zero at infinity in a mean square sense [2] lim R→+∞ ∂B If u is a D-solution to (1) 1 , then the traction field on the boundary is where a well defined field of W −1/2,2 (∂Ω) exists and the following generalized work and energy relation [1] holds where abuse of notation ∂Ω u · s(u) means the value of the functional s(u) ∈ W −1/2,2 (∂Ω) at u ∈ W 1/2,2 (∂Ω), and n is the unit outward (with respect to Ω) normal to ∂Ω.
If C is constant, then existence and regularity hold under the weak assumption of strong ellipticity [1], i.e., As far as we are aware, except for the property (7), little is known about the convergence at infinity of a D-solution and, in particular, whether or under what additional conditions (7) can be improved.
The main purpose of this paper is just to determine reasonable conditions on C assuring that (7) can be improved.
We say that C is regular at infinity if there is a constant elasticity tensor C 0 such that Let C 0 and C denote the linear spaces of the D-solutions to the equations for all τ ∈ R 3 and div C[∇h] = 0 in Ω, for all τ ∈ R 3 , A ∈ Lin, respectively.
The following theorem holds.
If C is regular at infinity, then

Preliminary Results
In this section, we collect the main tools we need to prove Theorem 1.
Moreover, if q = 2, then, for R R 0 , Proof. Lemma 1 is well-known (see, e.g., [2,4] and [5] Chapter II). We propose a simple proof for the sake of completeness. Since D 1,2 (Ω) is the completion of C ∞ 0 (Ω) with respect to the norm ∇u L 2 (Ω) , it is sufficient to prove (17) and (18) for a regular field u vanishing outside a ball. By basic calculus and Hölder inequality, Hence, (17) follows by a simple integration. From by Schwarz's inequality, one gets Hence, (18) follows at once.
Let C 0 be a constant and strongly elliptic elasticity tensor. The equation admits a fundamental solution U(x − y) [6] that enjoys the same qualitative properties as the well-known ones of homogeneous and isotropic elastostatics, defined by where µ is the shear modulus and ν the Poisson ratio (see [1] Section 51). In particular, U(x) = O(r −1 ) and for f with compact support (say) the volume potential is a solution (in a sense depending on the regularity of f ) to the system Let H 1 denote the Hardy space on R 3 (see [7] Chapter III). The following result is classical (see, e.g., [7]). Lemma 2. ∇ 2 V maps boundedly L q into itself for q ∈ (1, +∞) and H 1 into itself. and where C is the space of D-solutions to system (11).

Proof.
Let with R R 0 . Scalar multiplication of both sides of (1) 1 by gh, (2) and an integration by parts yield where e r = x/r. Since R ≤ |x| ≤ 2R, by Schwarz's inequality, Hence, letting R → +∞ and taking into account Lemma 1, (23) follows.

Lemma 4.
Let u be the D-solution to (1); then, for R R 0 , where C is the space of D-solutions to system (11).

Proof. (25) is easily obtained by integrating the identity
over B R and using the divergence theorem.

Proof of Theorem 1
Let ϑ(r) be a regular function, vanishing in B R and equal to 1 outside B 2R , for R R 0 . The field v = ϑu is a D-solution to the equation Of course, f ∈ L 2 (R 3 ) vanishes outside T R . Let C 0 be a strongly elliptic elasticity tensor. Clearly, v is a D-solution to the system which coincides with u outside B 2R . Since by Lemma 2, the map is continuous from D 1,q into itself, for q ∈ (3/2, +∞). Choose the map (30) is contractive in a neighborhood of 2 and its fixed point must coincide with v . Hence, there is q ∈ (1, 2) such that u ∈ D 1,q (Ω) and (12) is proved. If C is regular at infinity, then by Lemma 1 and the property of ϑ, Since the constant c(q) is uniformly bounded in every interval [a, b] and C − C 0 L ∞ ( S R 0 ) is sufficiently small, u ∈ D 1,q for q ∈ (3/2, +∞). Assume that ∂Ωû · s(h) = 0, ∀h ∈ C 0 .
By Lemma 3, for R R 0 , Therefore, taking into account (27), Since Then, by (31), the map (30) is contractive for q in a right neighborhood of 1 so that Conversely, if (35) holds, then a simple computation yields where g is the function (24). By Hölder's inequality, . Therefore, letting R → +∞ in (36) yields ∂Ω s(u) = 0 and this implies (32). From so that ∇V[ f ] ∈ L 1 . Since f ∈ L 2 (R 3 ) has compact support and satisfies (33), it belongs to H 1 (see [7] p. 92) and by Lemma 2 V[ f ] ∈ H 1 . Hence, it follows that (30) maps H 1 into itself and Since, by assumptions, C − C 0 L ∞ ( S R 0 ) is small, (30) is a contraction and by the above argument its (unique) fixed point must coincide with v so that ∇u ∈ L 1 (Ω).
We aim at concluding the paper with the following remarks.
(i) It is evident that the hypothesis that C is regular at infinity can be replaced by the weaker one that |C − C 0 | is suitably small at a large spatial distance. (ii) The operator V maps boundedly the Hardy space H q (q ∈ (0, 1]) into itself [7]. Hence, the argument in the proof of (16) can be used to show that ∇v ∈ H q , q > 3/4. We can then use the Sobolev-Poincaré (see [8] p. 255) to see that u ∈ L q (Ω) for q > 1.

Conflicts of Interest:
The author declares no conflict of interest.