Counter-examples to the Strichartz estimates for the wave equation in domains II

TL;DR

This paper constructs specific wave equation solutions in bounded domains that violate standard Strichartz estimates due to boundary-induced micro-local phenomena like caustics.

Contribution

It demonstrates the failure of Strichartz estimates in bounded domains by explicitly constructing counter-examples involving boundary effects.

Findings

Counter-examples violate Strichartz estimates in bounded domains.

Boundary effects generate caustics affecting wave behavior.

Results highlight limitations of free-space estimates in bounded settings.

Abstract

We consider a smooth and bounded domain of dimension d>1 and we construct solutions to the wave equation with Dirichlet boundary conditions which contradict the Strichartz estimates of the free space, at least for a subset of the usual range of indices. This is due to micro-local phenomena such as caustics generated in arbitrarily small time near the boundary.