Uniqueness in the Faber-Krahn Inequality for Robin Problems

Abstract

We prove uniqueness in the Faber-Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that amongst all sufficiently smooth domains of fixed volume, the ball is the unique minimiser for the first eigenvalue. The method of proof, which avoids the use of any symmetrisation, also works in the case of Dirichlet boundary conditions. We also give a characterisation of all symmetric elliptic operators in divergence form whose first eigenvalue is minimal.