Strichartz estimates for the Schr\"odinger equation on polygonal domains

TL;DR

This paper establishes Strichartz estimates with derivative loss for the Schrödinger equation on polygonal domains by relating them to Euclidean surfaces with conical singularities and using spectral analysis techniques.

Contribution

It introduces a method to derive Strichartz estimates on polygonal domains via doubling procedures and spectral analysis on conical surfaces, extending previous results to new geometric settings.

Findings

Strichartz estimates hold with derivative loss on polygonal domains.

A Littlewood-Paley squarefunction estimate is developed for these spaces.

Localization in space and time allows reduction to Euclidean cone estimates.

Abstract

We prove Strichartz estimates with a loss of derivatives for the Schr\"odinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on 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 result of the second author regarding the Schr\"odinger equation on the Euclidean cone.