On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

TL;DR

This paper analyzes the asymptotic behavior of Robin eigenvalues of the Laplacian in the exterior of convex polygons, showing they approach a quadratic function of the boundary parameter with corrections related to Dirichlet eigenvalues.

Contribution

It provides a precise asymptotic expansion for the Robin eigenvalues in polygonal exterior domains, linking them to Dirichlet eigenvalues of one-dimensional Laplacians.

Findings

Eigenvalues behave as -α^2 plus Dirichlet eigenvalues plus an error term as α→+∞

Asymptotic expansion connects Robin eigenvalues to boundary geometry

Results applicable to convex polygonal exterior domains

Abstract

Let $\Omega\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $\alpha>0$, let $H^\Omega_\alpha$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary conditions $\partial u/\partial\nu =\alpha u$, where $\nu$ is the exterior unit normal at the boundary of $\Omega$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^\Omega_m(\alpha)$ of $H^\Omega_\alpha$ behaves as \[ E^\Omega_m(\alpha)=-\alpha^2+\mu^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrt\alpha}\Big) \quad {as $\alpha$ tends to $+\infty$}, \] where $\mu^D_m$ stands for the $m$th eigenvalue of the operator $D_1\oplus...\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f"$ on $(0,\ell_n)$ with the Dirichlet boundary conditions.