A Faber-Krahn inequality for the Laplacian with Generalised Wentzell boundary conditions

Abstract.

We give an extension of the Faber-Krahn inequality to the Laplacian Δ on bounded Lipschitz domains $\Omega \subset {\mathbb{R}}^N, N \geq 2$, with generalised Wentzell boundary conditions $\Delta u + \beta\frac{\partial u}{\partial v} + \gamma u = 0$ on ∂Ω, where β, γ are nonzero real constants. We prove that when β, γ > 0, the ball B minimises the first eigenvalue with respect to all Lipschitz domains Ω of the same volume as B, and that B is the unique minimiser amongst C ²-domains. We also consider β, γ not both positive, and slightly extend what is known about the associated Wentzell operator and its resolvent in addition to considering an analogue of the Faber-Krahn inequality. This is based on the recent extension of the Faber-Krahn inequality to the Robin Laplacian. We also give a version of Cheeger’s inequality for the Wentzell Laplacian when β, γ > 0.