PDE Seminar Abstracts

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.

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