Spectral asymptotics for Robin Laplacians on polygonal domains

TL;DR

This paper analyzes the asymptotic behavior of the eigenvalues of Robin Laplacians on polygonal domains as the boundary parameter grows large, revealing that vertex models dominate the leading order and establishing Weyl asymptotics similar to smooth domains.

Contribution

It provides a detailed spectral asymptotic analysis for Robin Laplacians on polygonal domains, linking eigenvalue behavior to vertex model operators and extending Weyl law results.

Findings

First eigenvalues are determined by vertex model operators.

Eigenpairs are exponentially close to model operator eigenpairs in polygons with straight edges.

Weyl asymptotics for eigenvalue counting function match those of smooth domains.

Abstract

Let $\Omega$ be a curvilinear polygon and $Q^\gamma_{\Omega}$ be the Laplacian in $L^2(\Omega)$, $Q^\gamma_{\Omega}\psi=-\Delta \psi$, with the Robin boundary condition $\partial_\nu \psi=\gamma \psi$, where $\partial_\nu$ is the outer normal derivative and $\gamma>0$. We are interested in the behavior of the eigenvalues of $Q^\gamma_\Omega$ as $\gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^\gamma_\Omega$ is determined at the leading order by those of model operators associated with the vertices: the Robin Laplacians acting on the tangent sectors associated with $\partial \Omega$. In the particular case of a polygon with straight edges the first eigenpairs are exponentially close to those of the model operators. Finally, we prove a Weyl asymptotics for the eigenvalue counting function of $Q^\gamma_\Omega$ for a threshold depending on $\gamma$, and show that the leading term is the same as for smooth domains.