Strichartz estimates for Dirichlet-wave equations in two dimensions with applications

TL;DR

This paper proves the Strauss conjecture for wave equations with obstacles in two dimensions by establishing Strichartz estimates, overcoming previous difficulties related to Sobolev index requirements through interpolation techniques.

Contribution

It introduces a novel interpolation method to derive Strichartz estimates in 2D obstacle problems, enabling proof of the Strauss conjecture in this challenging setting.

Findings

Established Strichartz estimates for 2D obstacle wave equations.

Proved the Strauss conjecture for nontrapping obstacles in two dimensions.

Developed interpolation techniques between energy and Minkowski space estimates.

Abstract

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out in \cite{HMSSZ} this case is more subtle than $n=3$ or 4 due to the fact that the arguments of the first two authors \cite{SmSo00}, Burq \cite{B} and Metcalfe \cite{M} showing that local Strichartz estimates for obstactles imply global ones require that the Sobolev index, $\gamma$, equal 1/2 when $n=2$. We overcome this difficulty by interpolating between energy estimates ($\gamma =0$) and ones for $\gamma=\frac12$ that are generalizations of Minkowski space estimates of Fang and the third author \cite{FaWa2}, \cite{FaWa}, the second author \cite{So08} and Sterbenz \cite{St05}.