Dispersive estimates for the semi-classical Schrödinger equation inside a strictly convex domain, preprint
Dispersive estimates inside the Friedlander model domain for the Klein-Gordon equation and the wave equation in large time, preprint
- Preprint : with Richard Lascar, Gilles Lebeau and Fabrice Planchon
- 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 tˆ¼ 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.
- 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 Rˆ³. For d ≥4 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.
This project has received funding from the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757 996)