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 $\Omega$ 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 $L_t^4{L}_x^{\infty }$ 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.

Title change, and to appear in Math Annalen

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• Riesz transforms outside a convex obstacle
• Global well-posedness and scattering for defocusing energy-critical NLS in the exterior of balls with radial data
• Strichartz estimates for the Schrödinger equation for the sublaplacian on complex spheres