How to hear the corners of a drum

TL;DR

This paper proves that the corners of a planar domain are a spectral invariant of the Laplacian under various boundary conditions, using a locality principle and heat kernel calculations, extending previous results to Robin and Neumann cases.

Contribution

It introduces a new locality principle applicable to all three boundary conditions and extends spectral invariance of corners to Robin and Neumann cases, previously known only for Dirichlet.

Findings

Corners are spectral invariants under Dirichlet, Neumann, and Robin conditions.

A locality principle is established for all three boundary conditions.

Microlocal analysis methods are used for curvilinear polygons.

Abstract

We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.