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.

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

References in corpus (8)

• L^2-torsion of hyperbolic manifolds of finite volume
• A heat trace anomaly on polygons
• One can hear the corners of a drum
• Semigroup Properties for the Second Fundamental Form
• The sound of symmetry
• A Polyakov formula for sectors
• The Neumann isospectral problem for trapezoids
• Gradient Estimate on the Neumann Semigroup and Applications

Cited by in corpus (2)

• The heat kernel on curvilinear polygonal domains in surfaces
• Polyakov formulas for conical singularities in two dimensions