ON THE FIRST NONTRIVIAL EIGENVALUE OF THE ∞LAPLACIAN WITH NEUMANN BOUNDARY CONDITIONS

We study the limit as p -> infinity of the first non-zero eigenvalue gimel(p) of the p-Laplacian with Neumann boundary conditions in a smooth bounded domain U subset of R-n. We prove that gimel(infinity) := lim(p) ->+infinity gimel(1/p)(p) = 2/diam(U), where diams (U) denotes the diameter of U with respect to the geodesic distance in U. We can think of gimel(infinity) as the first eigenvalue of the infinity-Laplacian with Neumann boundary conditions. We also study the regularity of gimel(infinity) as a function of the domain U proving that under a smooth perturbation U-t of U by diffeomorphisms close to the identity there holds that gimel(infinity)(U-t) = (U)+ O(t). Although gimel(infinity)(U-t) is in general not differentiable at t = 0, we show that in some cases it is so with an explicit formula for the derivative.