Dispersion for the wave equation inside strictly convex domains II  
preprint ; with Fabrice Planchon, Gilles Lebeau and Richard Lascar 
In this  work, we consider the wave equation on a generic strictly convex domain of dimension at least two with smooth non empty 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 t^{1/4} loss with respect to the boundary less case. Moreover, we precisely describe where and when these losses occur and relate them to swallowtail type singularities in the wave front set, proving that the resulting decay is optimal.
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 be a powerful tool to prove sharp propagation of singularities results which were out of reach until now.
Dispersion estimates for wave and Schrödinger equations outside a strictly convex obstacles and counterexamples
preprint in collaboration with Gilles lebeau 
Abstract: We consider the linear wave equation and the linear Schrödinger equation outside a compact, strictly convex obstacle in R^d with smooth boundary. In dimension d = 3 we show that the linear flow satisfies the dispersive estimates as in R3. For d >3 and if the obstacle is a ball, we show that there exists points (near the Poisson-Arago spot) where the dispersive estimates fail for both wave and Schrödinger quations.
The question about whether or not dispersion did hold outside general strictly convex obstacles was raised more than 20 years ago : in this work we give sharp answers which highlight the importance of diffractive effects, especially in higher dimensions where we provide unexpected counterexamples.