Strichartz estimates for the wave equation on manifolds with boundary

Abstract

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

Strichartz estimates and nonlinear wave equation on nontrapping asymptotically conic manifolds

“…See for examples [21,26,32]. Strichartz estimates also are considered on compact manifold with boundary, see [6], [2] and references therein. When we consider the noncompact manifold with nontrapping condition, one can obtain global-in-time Strichartz estimates.…”

Dispersive and Strichartz estimates for the wave equation in domains with boundary

“…The restrictions on the indices in [1] are naturally imposed by the local nature of the paramerix construction in [19]. In dimension d ≥ 3 the result is certainly not optimal (Lebeau's applications give larger sets of admissible indices), and this is a consequence of the fact that in higher dimensions the approach in [1] does not allow to describe the dispersion effect in the d − 2 tangential variables.…”

“…The main observation in [1] is that one can construct parametrices over large time intervals when moving to directions which are not tangential to ∂Ω. Precisely, for directions of angle θ one can construct a parametrix on intervals of time of size θ, yielding to a θ-depending loss in the Strichartz estimates.…”

“…In [3], N.Burq, G.Lebeau and F.Planchon were able to use the square function estimates from [19] to prove Strichartz estimates without loss for solutions of (1.1) for the admissible pair (d = 3, q = 5, r = 5) that allowed them to show that there is global existence for the H 1 -critical nonlinear wave equation for domains in R 3 . Shortly after this, M.Blair, H.Smith and C.Sogge obtained in [1] optimal Strichartz estimates for triples which satisfy:…”

“…The strategy in [1] consists in doubling the manifold Ω along its boundary to produce a boundaryless manifold with a special type of Lipschitz metric (with codimension-1 singulairities). The study of the general wave equation for general Lipschitz metrics has already been developed by H.Smith [17] and D.Tataru [21] by introducing new wave-packet techniques and is particularly useful in the study of eigenfunctions estimates (a recent example is give by the work [19]).…”