The elastic echo problem

Abstract

SUMMARYWe consider a homogeneous isotropic unbounded linear elastic medium ⊂ R 3 , having a free boundary . A forcing f(t; x) creates an incident displacement ÿeld u 0 (t; x). This primary ÿeld is scattered by giving rise to a secondary ÿeld or echo, for which we determine the asymptotic behaviour in time. NOTATIONSLet us ÿrst explain the notations which are used throughout the text. Some of them are deÿned more precisely in the appendix. We recommend to the reader which is not familiarized with the scattering theory of Wilcox to readÿrst this appendix.The space R 3 is endowed with the canonical basis {e 1 = (1; 0; 0); e 2 = (0; 1; 0); e 3 = (0; 0; 1), the origin O and the co-ordinate system (x 1 ; x 2 ; x 3 ). The canonical scalar product of R N is denoted by a dot. I is the identity map. For a matrix or a map A, For any vector u ∈ C 1 ( ), we deÿne the gradient tensor ∇u(x) and the linearized deformation tensor (u) = 1 2 (∇U + t ∇u) with components ij (u) = 1 2 (@u i =@x j + @u j =@x i ) and (u) will be the linearized stress tensor of u. By (: ; :), we denote the scalar product in L 2 . We deÿne also the following Hilbert spaces:2 ( * ; ) {u; u satisfait (NG)} where (NG) is the generalized Neumann condition (NG):We use also the localized counterparts (Frà echet spaces) L ; L N; loc ; L loc 2 ( * ; ) for functions which are locally in the mentioned spaces.Let X a Hilbert space and I ⊆ R. Then C(I; X ) is the Banach space of continuous X -valued functions and H 1 (I; X ) is the Hilbert space of X -valued functions, which are square integrable together with their ÿrst derivative on I .∇ is the usual di erential operator. If u is a scalar function, ∇u is a vector ( the gradient), ∇ 2 u = ∇∇u is the Hessian matrix of u, with elements @ 2 u=@x i x j , and ∇u is a matrix if u is a vectorial function.A 0 (resp., A) is the abstract operator associated with the free space (resp. exterior ) problem (see the appendix). For an operator T , D(T ) denotes the domain, T * the adjoint and T −1 the inverse. is the Fourier transform, * its adjoint (and inverse). The dual variable of x is p. Thus if u(x) ∈ L 2 (R 3 );û(p) = u(p) is its Fourier transform. If (·) is some expression, we shall also denote sometimes by (·) ∧ its Fourier transform.grad div is the Lamà e operator, ; are the Lamà e constants. T(n; u) = (u)n is the traction vector at the point x of the boundary where the normal is n. [L 2 (R 3 )] 2 respectively)P 1 = Á ⊗ Á,P 2 = I −Á ⊗ Á; Á = p=|p|. We deÿne then j =P j ;is the Green's function of the free space, unique solution of the problem ( * x + z)G 0 (x; x ; z) = − (x − x )I; ∀x; x ∈ R 3 . For real positive , we deÿne the two matrices G ± 0 (x; x ; ) = G 0 (x; x ; ± i0) by the limiting absorption technique: