Spectral analysis of a perturbed multistratified isotropic elastic strip: New method

We study the self-adjoint operator (D(A), A) associated with an elastic iso tropic and multistratified strip Omega = {(x(1), x(2)) is an element of R-2 ; 0 < x(2) < L}, which means that there exists a constant a > 0 such that t he density rho and Lame coefficients lambda and mu are, for (- 1)(k)x(1) gr eater than or equal to a, k = 1,2, respectively, equal to functions rho(k), lambda(k) and mu(k), depending only on x(2). Thanks to [4] the properties of the free operators A(k), k = 1,2, associated with rho(k), lambda(k) and mu(k), are well-known. We study A by considering it as a 'compact perturbat ion' of the pair (A(1), A(2)). The difficulty is: if psi is an element of C -0(infinity)(R-2) and u is an element of D(A) then psi u does not necessari ly belong to D(A). It has already been encountered in other studies concern ing elasticity (cf. [10,18]). Adapting the techniques used there to overcom e this difficulty imposes restrictive conditions on lambda(k) and mu(k). Th e purpose of this paper is to propose a new method, which removes definitiv ely this difficulty and enables us without restrictive conditions on lambda (k) and mu(k) to prove a limiting absorption principle for A. Copyright (C) 1999 John Wiley & Sons, Ltd.