On wave equation outside trapping obstacles and local energy decay in odd space dimensions

TL;DR

This paper establishes resolvent estimates for the Dirichlet Laplacian on exterior domains with general geometries, leading to local energy decay results for wave equations in odd dimensions without requiring non-trapping conditions.

Contribution

It introduces new resolvent estimates under broad geometric assumptions, enabling local energy decay analysis without non-trapping constraints.

Findings

Resolvent estimates hold without non-trapping assumptions.

Local energy decay is proven for wave equations in odd dimensions.

Results apply to general exterior domains with minimal geometric restrictions.

Abstract

The purpose of the present paper is to establish appropriate cut-off resolvent estimates for the Dirichlet Laplacian on exterior domains. The geometrical assumptions on domains are rather general, for example, non-trapping condition is not imposed. The first key assumption guarantees the result on propagation of singularities, the second is the smallness of the Lebesgue measure of the portions of the trapped sets in the fibers of cosphere bundle, and the third concerns the upper bound of the sojourn time. As a by-product of these estimates, the local energy decay estimate for solutions to the initial-boundary value problem for wave equation in the case of odd space dimensions is obtained.