Spectral asymptotics for Robin Laplacians on polygonal domains

Abstract

Let $Ω$ be a curvilinear polygon and $Q^γ_Ω$ be the Laplacian in $L^2(Ω)$, $Q^γ_Ωψ=-Δψ$, with the Robin boundary condition $\partial_νψ=γψ$, where $\partial_ν$ is the outer normal derivative and $γ>0$. We are interested in the behavior of the eigenvalues of $Q^γ_Ω$ as $γ$ becomes large. We prove that the asymptotics of the first eigenvalues of $Q^γ_Ω$ 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 $\partial Ω$. 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 $Q^γ_Ω$ for a threshold depending on $γ$, and show that the leading term is the same as for smooth domains.