The orthotropic $p$-Laplace eigenvalue problem of Steklov type as $p\to +\infty$

Abstract

We study the Steklov eigenvalue problem for the $\infty -$orthotropic Laplace operator defined on convex sets of ${\mathbb{R}}^N$, with $N\ge 2$, considering the limit for $p\to +\infty$ of the Steklov problem for the $p-$orthotropic Laplacian. We find a limit problem that is satisfied in the viscosity sense and a geometric characterization of the first non trivial eigenvalue. Moreover, we prove Brock-Weinstock and Weinstock type inequalities among convex sets, stating that the ball in a suitable norm maximizes the first non trivial eigenvalue for the Steklov $\infty -$orthotropic Laplacian, once we fix the volume or the anisotropic perimeter.

30 pages