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

Abstract:

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension $n$ is two. As pointed out by Hidano, Metcalfe, Smith, Sogge, and Zhou (2010) this case is more subtle than $n=3$ or $4$ due to the fact that the arguments in the papers of the first two authors (2000), Burq (2000) and Metcalfe (2004), showing that local Strichartz estimates for obstacles imply global ones, require that the Sobolev index, $\gamma$, equals $1/2$ when $n=2$. We overcome this difficulty by interpolating between energy estimates ($\gamma =0$) and ones for $\gamma =\frac 12$ that are generalizations of Minkowski space estimates of Fang and the third author (2006), (2011), the second author (2008) and Sterbenz (2005).