Dispersion for the wave equation inside strictly convex domains II: the general case

TL;DR

This paper develops sharp dispersion estimates for the wave equation in strictly convex domains, revealing precise loss mechanisms and establishing optimal decay rates with improved Strichartz estimates.

Contribution

It constructs a sharp local parametrix and characterizes the boundary-induced dispersion loss, providing new insights into wave behavior in convex geometries.

Findings

Decay rate exhibits a $t^{1/4}$ loss compared to boundaryless case.

Losses are linked to swallowtail singularities in wave front set.

Strichartz estimates are improved, showing minimal average decay loss.

Abstract

We consider the wave equation on a manifold $(\Omega,g)$ of dimension $d\geq 2$ with smooth strictly convex boundary $\partial\Omega\neq\emptyset$, with Dirichlet boundary conditions. We construct a sharp local in time parametrix and then proceed to obtain dispersion estimates: our fixed time decay rate for the Green function exhibits a $t^{1/4}$ loss with respect to the boundary less case. We precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that our decay is optimal. Moreover, we derive better than expected Strichartz estimates, balancing lossy long time estimates at a given incidence with short time ones with no loss: for $d=3$, it heuristically means that, on average the decay loss is only $t^{1/6}$.