Strichartz and Localized Energy Estimates for the Wave Equation in Strictly Concave Domains

TL;DR

This paper establishes localized energy and Strichartz estimates for the wave equation in strictly concave domains with Dirichlet or Neumann boundary conditions, extending techniques to Neumann cases and Schrödinger equations.

Contribution

It introduces a new approach for deriving localized energy and Strichartz estimates in concave domains, applicable to both Dirichlet and Neumann boundary conditions, and extends to Schrödinger equations.

Findings

Strichartz estimates are obtained for wave equations with Neumann boundary conditions.

Localized energy estimates show a gain in regularity near the boundary.

The method applies to Schrödinger equations with time-independent coefficients.

Abstract

We prove localized energy estimates for the wave equation in domains with a strictly concave boundary when homogeneous Dirichlet or Neumann conditions are imposed. By restricting the solution to small, frequency dependent, space time collars of the boundary, it is seen that a stronger gain in regularity can be obtained relative to the usual energy estimates. Mixed norm estimates of Strichartz and square function type follow as a result, using the energy estimates to control error terms which arise in a wave packet parametrix construction. While the latter estimates are not new for Dirichlet conditions, the present approach provides an avenue for treating these estimates when Neumann conditions are imposed. The method also treats Schr\"odinger equations with time independent coefficients.