Non-Trapping Conditions and Local Energy Decay for Hyperbolic Problems

Abstract

Uniform local energy decay for solutions of hyperbolic problems in exterior domains is only possible under certain assumptions on the obstacle and the medium. The most general of these assumptions are the so-called "non-trapping conditions" and the literature provides several of them. We compare these conditions and give some geometric interpretation, dealing with boundary value problems for the wave equation and the equations of isotropic elasticity in both homogeneous and inhomogeneous media. In establishing our results we prove in addition the existence and uniqueness of the Green's function (Green's matrix) of the above problems.