Spectral asymptotics for Robin Laplacians on polygonal domains

Abstract

Let $\Omega$ be a curvilinear polygon and $Q_{\Omega }^{\gamma }$ be the Laplacian in $L^2(\Omega )$, $Q_{\Omega }^{\gamma }\psi =-\Delta \psi$, 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 $Q_{\Omega }^{\gamma }$ as $\gamma$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q_{\Omega }^{\gamma }$ 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 $Q_{\Omega }^{\gamma }$ for a threshold depending on $\gamma$, and show that the leading term is the same as for smooth domains.

38 pages