Spectral asymptotics for Robin Laplacians on polygonal 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.