Strichartz Estimates for Dirichlet-wave Equations in Two Dimensions with Applications

We establish the Strauss conjecture for nontrapping obstacles when the spatial dimension n is two. As pointed out in [7] this case is more subtle than n = 3 or 4 due to the fact that the arguments of the first two authors [11], Burq [1] and Metcalfe [9] showing that local Strichartz estimates for obstacles imply global ones require that the Sobolev index, γ, equal 1/2 when n = 2. We overcome this difficulty by interpolating between energy estimates (γ = 0) and ones for γ = 1 2 that are generalizations of Minkowski space estimates of Fang and the third author [4], [5], the second author [12] and Sterbenz [14].