Some overdetermined problems related to the anisotropic capacity

TL;DR

This paper characterizes the Wulff shape of an anisotropic norm through overdetermined boundary value problems involving the Finsler p-capacity, revealing geometric conditions linked to the shape of the domain.

Contribution

It establishes a characterization of the Wulff shape via overdetermined problems for the Finsler p-capacity, connecting shape properties to potential level set behavior.

Findings

If the Finsler p-capacitary potential has two homothetic level sets, then the domain is Wulff shape.

The concavity exponent of the potential is - (p-1)/(N-p) if and only if the domain is Wulff shape.

The paper links geometric shape characterization to solutions of overdetermined boundary problems.

Abstract

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler $p$-capacity of a convex set $\Omega \subset \mathbb{R}^N$, with $1<p<N$. In particular we show that if the Finsler $p$-capacitary potential $u$ associated to $\Omega$ has two homothetic level sets then $\Omega$ is Wulff shape. Moreover, we show that the concavity exponent of $u$ is $q=-(p-1)/(N-p)$ if and only if $\Omega$ is Wulff shape.