Local energy decay for Lipschitz wavespeeds

TL;DR

This paper establishes a logarithmic decay rate for wave energy in media with Lipschitz continuous wavespeeds, extending previous results to less smooth perturbations.

Contribution

It proves decay rates for wave equations with Lipschitz wavespeeds using resolvent estimates, generalizing prior smooth perturbation results.

Findings

Logarithmic local energy decay rate proven

Resolvent estimates at high and low energies established

Decay rate matches that for smooth perturbations

Abstract

We prove a logarithmic local energy decay rate for the wave equation with a wavespeed that is a compactly supported Lipschitz perturbation of unity. The key is to establish suitable resolvent estimates at high and low energy for the meromorphic continuation of the cutoff resolvent. The decay rate is the same as that proved by Burq for a smooth perturbation of the Laplacian outside an obstacle.