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.