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.
Keywords
Cite
@article{arxiv.2012.03366,
title = {How to hear the corners of a drum},
author = {Medet Nursultanov and Julie Rowlett and David Sher},
journal= {arXiv preprint arXiv:2012.03366},
year = {2020}
}
Comments
This is the author original manuscript that was submitted for publication in Matrix Annals. The published version has been revised