Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary

Abstract. We establish Strichartz estimates for the Schrödinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key L4 t L∞x estimate, which we use to give a simple proof of well-posedness results for the energy critical Schrödinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schrödinger equations on general compact manifolds with boundary. 1