On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions
Julio Daniel Rossi, Nicolas Saintier · Americanae (AECID Library) · 2016
Abstract. We study the limit as p→ ∞ of the first non-zero eigenvalue λp of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U ⊂ Rn. We prove that λ ∞: = limp→+ ∞ λ1/pp = 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 ∞-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 prove that in some cases it is so with an explicit formula for the derivative. 1. introduction Denote by λp the first non-zero eigenvalue of the p-Laplacian with Neu-mann boundary conditions in a smooth bounded domain U ⊂ Rn. The aim of this paper is two-fold. We first study the asymptotic behaviour of λp as p→∞, obtaining that λ ∞: = lim p→+∞λ