Absence of Positive Eigenvalues for the Linearized Elasticity System

In this paper, we prove that the linearized elasticity system has no eigenvalues in two geometric situations: the whole space and a local perturbation of the half-space. We consider the Lamé coefficients and the density varying in an unbounded part of the domain. For the whole space, we use the operations curl and div to reduce our system to a scalar problem and use a limiting absorption principle for the reduced scalar equation given by the partial Fourier transform. For the perturbed half-space, this decompositions being no longer valid, we give an other method based on a pseudo-decomposition using the operations div and curl in the horizontal direction. In contrast to the whole space case, the reduced problems depend strongly on the dual Fourier variable which do not enable us to use same techniques. To study these reduced problems, we use the analytic theory of linear operators.