The asymptotic behaviour of the eigenvalues of a Robin problem

Abstract

We consider the eigenvalues of the Laplacian with Robin-type boundary conditions $\frac{\partial u}{\partial \nu }=\alpha u$. Here we assume $\alpha >0$, in contrast to the usual case where $\alpha <0$. In recent years, increasing attention has been devoted to the behaviour of the smallest eigenvalue ${\lambda }_1$ as the parameter $\alpha \to \infty$ under various assumptions on the underlying domain. After surveying existing results in this area, we will prove using a test function argument that every eigenvalue ${\lambda }_n$ has the same asymptotic behaviour, ${\lambda }_n~-{\alpha }^2$, assuming only that $\Omega$ is of class $C^1$. This is joint work with Daniel Daners.