Steklov eigenvalues for the $\infty$-Laplacian

Abstract

We study the Steklov eigenvalue problem for the $\infty$-laplacian. To this end we consider the limit as $p\to \infty$ of solutions of $-{\Delta }_p{u}_p=0$ in a domain $\Omega$ with $|\nabla u_p{|}^{p-2}\partial u_p/\partial \nu =\lambda |u{|}^{p-2}u$ on $\partial \Omega$. We obtain a limit problem that is satisfied in the viscosity sense and a geometric characterization of the second eigenvalue.