Dispersion for the wave and Schr\"odinger equations outside a ball and counterexamples

Abstract

We consider the wave equation with Dirichlet boundary conditions in the exterior of the unit ball $B_{d}(0,1)$ of $\mathbb{R}^d$. For $d=3$, we obtain a global in time parametrix and derive sharp dispersive estimates, matching the $\mathbb{R}^{3}$ case, for all frequencies (low and high). For $d\geq 4$, we provide an explicit solution at large frequency $1/h$, $h\in (0,1)$, with a smoothed Dirac data at a point at distance $h^{-1/3}$ from the origin in $\mathbb{R}^d$ whose decay rate exhibits $h^{-(d-3)/3}$ loss with respect to the boundary less case, that occurs at observation points around the mirror image of the source with respect to the center of the ball (at the Poisson-Arago spot). Similar counterexample are obtained for the Schr{\"o}dinger flow. Moreover, we generalize these counterexamples, first announced in \cite{ildispext}, to the case of the wave and Schr{\"o}dinger equations outside cylindrical domains of the form $B\_{d\_1}(0,1)\times \mathbb{R}^{d\_2}$ in $\mathbb{R}^d$ with $d=d\_1+d\_2$ and $d\_1\geq 4$, for which we construct solutions, as done \cite{IaIv23} for $d\_1=2$, $d\_2=1$, whose decay rates exhibit a $h^{-(d\_1-3)/3}$ loss with respect to the boundary less case (at observation points around the mirror image of the source with respect to the origin).