We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcricital case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.Comment: 16 pages. Couple of typos corrected, to appear in Annales de l'Institut Henri Poincar

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 L 4 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.

This article establishes the almost global existence of solutions of three-dimensional quadratically semilinear wave equations with the use of only the classical invariance of the equations under translations and spatial rotation. Using these techniques we can handle semilinear wave equations in Minkowski space or semilinear Dirichlet-wave equations in the exterior of a nontrapping obstacle.Our results and approach are related to previous work in the non-obstacle case. In particular, in [2], almost global existence was shown for general, quadratic quasilinear wave equations in Minkowski space, using fully the invariance properties of the wave operator under translations, spatial rotations, scaling, and Lorentz boosts . Subsequently, in [5], the almost global existence of certain divergence-form quadratic quasilinear wave equations was established in Minkowski space, using only the invariance of the wave operator under translations, spatial rotations, and changes of scale. Further techniques and results concerning equations from the theory of elasticity were presented in [9], [10].The main difference between our approach and earlier work is that it only exploits the O(|x| −1 ) decay of solutions of wave equations with compactly supported data, as opposed to the stronger O(t −1 ) decay. Here of course x = (x 1 , x 2 , x 3 ) is the spatial component and t the time component of a space-time vector (t, x) ∈ R + × R 3 . Establishing O(|x| −1 ) decay is considerably easier and can be achieved using only the invariance with respect to translations and spatial rotations. Fortunately, as we shall see, one can set up an iteration argument that only requires this weaker decay to obtain almost global existence. Another advantage to this approach for obstacle problems is that the translation and spatial rotational vector fields essentially preserve the Dirichlet conditions. The homogeneous vector fields used in [2] do not have this property for any obstacle, and consequently it seems difficult to use them for nonlinear obstacle problems.