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

TL;DR

This paper investigates the limit behavior of Steklov eigenvalues for the orthotropic $p$-Laplace operator as $p$ approaches infinity, providing a geometric characterization and inequalities for convex sets.

Contribution

It introduces a limit problem for the Steklov eigenvalues of the $ abla_ ext{orthotropic}^ ext{infty}$ operator and establishes extremal inequalities among convex sets.

Findings

Limit problem characterized in viscosity sense.

Ball maximizes the first eigenvalue under volume or perimeter constraints.

Geometric characterization of the first non trivial eigenvalue.

Abstract

We study the Steklov eigenvalue problem for the $\infty-$orthotropic Laplace operator defined on convex sets of $\mathbb{R}^N$, with $N\geq2$, 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.