Abstract

In recent years, following results on dispersive estimates for low regularity metrics, substantial progress has been made on dispersive estimates for the wave and Schrodinger equations on domains. Here we report on recent work to obtain a sharp dispersion estimate. For this, we rely on a precise description of the wave front (or the pseudo-spheres, e.g. surfaces reached by light emanating from a point after a fixed amount of time) and on a suitable microlocal parametrix construction near the boundary, for the wave equation inside strictly convex domains, subject to Dirichlet boundary condition. Such a parametrix allows to follow wave packets propagating along the boundary with a large number of reflections. In the process we encounter Fourier Integral Operators whose canonical forms correspond to cusp and swallowtail singularities, and which account for the loss (compared to the boundary less case) in dispersive estimates. This is joint work with Gilles Lebeau and Fabrice Planchon.