Dispersion for the wave equation inside strictly convex domains I: the Friedlander model case

Abstract

We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$, and we describe dispersion for the wave equation with Dirichlet boundary conditions. More specifically, we obtain the optimal fixed time decay rate for the smoothed out Green function: a $t^{1/4}$ loss occurs with respect to the boundary less case, due to repeated occurrences of swallowtail type singularities in the wave front set.

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Authors

Oana Ivanovici

CNRS and Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France

Gilles Lebeau

Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France, and Institut Universitaire de France

Fabrice Planchon

Université Nice Sophia-Antipolis, 06108 Nice Cedex 02, France and Institut Universitaire de France