Strichartz estimates for the Schrödinger equation on polygonal domains

Details Strichartz estimates for the Schrödinger equation on polygonal domains Strichartz estimates for the Schrödinger equation on polygonal domains

2010-03-02

arxiv.org/abs/1003.0666v1

J. Geom. Anal. (2012), Vol. 22, No. 2, 339-351

Mathematics

Analysis of PDEs

12 pages

Scientific paper

10.1007/s12220-010-9187-3

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.

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