PDE Seminar Abstracts

# The asymptotic behaviour of the eigenvalues of a Robin problem

James Kennedy
University of Sydney
10 May 2010 3-4pm, Carslaw Room 273

## 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.