On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions

Abstract

We study the limit as p goes to infinity of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U of Rn. We prove that λ∞:=lim λp1/p=2/diam(U), where diam(U) denotes the diameter of U with respect to the geodesic distance in U. We can think of λ∞ as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of λ∞ as a function of the domain U proving that, under a smooth perturbation Ut of U by diffeomorphisms close to the identity, there holds that λ∞(Ut)=λ∞(U)+O(t). Although λ∞(Ut) is in general not differentiable at t=0, we provide sufficient geometric conditions for its differentiability with an explicit formula for the derivative.