The Neumann eigenvalue problem for the $\infty$-Laplacian

Abstract

The first nontrivial eigenfunction of the Neumann eigenvalue problem for the $p$-Laplacian, suitably normalized, converges to a viscosity solution of an eigenvalue problem for the $\infty$-Laplacian as $p\to \infty$. We show among other things that the limiting eigenvalue, at least for convex sets, is in fact the first nonzero eigenvalue of the limiting problem. We then derive a number of consequences, which are nonlinear analogues of well-known inequalities for the linear (2-)Laplacian.