Counterexamples to the Strichartz inequalities for the wave equation in general domains with boundary

Abstract

In this paper we consider a smooth and bounded domain $\Omega \subset {\mathbb{R}}^d$ of dimension $d\ge 2$ with boundary and we construct sequences of solutions to the wave equation with Dirichlet boundary condition which contradict the Strichartz estimates of the free space, providing losses of derivatives at least for a subset of the usual range of indices. This is due to microlocal phenomena such as caustics generated in arbitrarily small time near the boundary. Moreover, the result holds for microlocally strictly convex domains in ${\mathbb{R}}^d$.