The \infty eigenvalue problem and a problem of optimal transportation

Abstract

The so-called eigenvalues and eigenfunctions of the infinite Laplacian ${\Delta }_{\infty }$ are defined through an asymptotic study of that of the usual $p$-Laplacian ${\Delta }_p$, this brings to a characterization via a non-linear eigenvalue problem for a PDE satisfied in the viscosity sense. In this paper, we obtain an other characterization of the first eigenvalue via a problem of optimal transportation, and recover properties of the first eigenvalue and corresponding positive eigenfunctions.