Spectral asymptotics for Robin Laplacians on polygonal domains

Abstract

Let Ω be a curvilinear polygon and QΩγ be the Laplacian in L2(Ω), QΩγψ=−Δψ, with the Robin boundary condition ∂νψ=γψ, where ∂ν is the outer normal derivative and γ>0. We are interested in the behavior of the eigenvalues of QΩγ as γ becomes large. We prove that the asymptotics of the first eigenvalues of QγΩ 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 ∂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 QΩγ for a threshold depending on γ, and show that the leading term is the same as for smooth domains.