Dispersion estimates for the wave and the Schrödinger equations outside a ball and counterexamples
Abstract : We consider the wave and the Schrödinger equations with Dirichlet boundary conditions in the exterior of a ball in Rd. In dimension d=3 we construct a sharp, global in time parametrix and then proceed to obtain sharp dispersive estimates, matching the R3 case, for all frequencies (low and high). If d ≥4 we provide an explicit solution to the wave equation localized at large frequency 1/h with data a Dirac mass at a point Q_0at large distance s from the center of the ball : taking s~1/h1/3, the decay rate of that solution exhibits a (t/h)(d-3)/4 loss with respect to the boundary less case, that occurs at t~ 2s with an observation point being symmetric to Q_0 with respect to the center of the ball (at the Poisson Arago spot). A similar counterexample is also obtained for the Schrödinger flow.
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.