Abstract interior convex domain

Dispersion for the wave equation inside strictly convex domains II

Abstract : In this paper, we consider the wave equation on a strictly convex domain Ω of dimension d ≥ 2 with smooth boundary and 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 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 1/6.

 The new methods in this paper (compared to the Friedlander model case) provide a much deeper understanding of the complex propagation pattern near the boundary and extends the parametrix construction to the largest possible phase space region. This may have far reaching consequences, beyond pointwise bounds, as the parametrix will prove to bea powerful tool to prove sharp propagation of singularities results which were out of reach until now.