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

Abstract

Let $Ω\subset \mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,...,\ell_M$. For $α>0$, let $H^Ω_α$ denote the Laplacian in $Ω$, $u\mapsto -Δu$, with the Robin boundary conditions $\partial u/\partialν=αu$, where $ν$ is the exterior unit normal at the boundary of $Ω$. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E^Ω_m(α)$ of $H^Ω_α$ behaves as \[ E^Ω_m(α)=-α^2+μ^D_m +\mathcal{O}\Big(\dfrac{1}{\sqrtα}\Big) \quad {as $α$ tends to $+\infty$}, \] where $μ^D_m$ 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.