How to hear the corners of a drum

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.

This is the author original manuscript that was submitted for publication in Matrix Annals. The published version has been revised

References in corpus (4)

• One can hear the corners of a drum
• Semigroup Properties for the Second Fundamental Form
• The sound of symmetry
• Gradient Estimate on the Neumann Semigroup and Applications