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.

Abstract. We prove local Strichartz estimates with a loss of derivatives over compact manifolds with boundary. Our results also apply more generally to compact manifolds with Lipschitz metrics.

We obtain quasimode, eigenfunction and spectral projection bounds for Schrödinger operators, H V = −∆g + V (x), on compact Riemannian manifolds (M, g) of dimension n ≥ 2, which extend the results of the third author [40] corresponding to the case where V ≡ 0. We are able to handle critically singular potentials and consequently assume that V ∈ L n 2 (M ) and/or V ∈ K(M ) (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where V ≡ 0 that go back to the third author [40] as well as ones which arose in the work of Kenig, Ruiz and this author [25] in the study of "uniform Sobolev estimates" in R n . We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural L p → L p spectral multiplier theorems under the assumption that V ∈ L n 2 (M ) ∩ K(M ). Moreover, we can also obtain natural analogs of the original Strichartz estimates [49] for solutions of (∂ 2 t −∆+V )u = 0. We also are able to obtain analogous results in R n and state some global problems that seem related to works on absence of embedded eigenvalues for Schrödinger operators in R n (e.g., [21], [22], [26], [27] and [34].)

Abstract. We prove Strichartz estimates with a loss of derivatives for the Schrödinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates on the polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a recent result of the second author regarding the Schrödinger equation on the Euclidean cone.