An exterior overdetermined problem for Finsler $N$-laplacian in convex cones

TL;DR

This paper proves a rigidity result for an overdetermined anisotropic N-Laplace problem in convex cones, showing solutions imply the domain is a Wulff shape intersection, using Pohozaev identities and isoperimetric inequalities.

Contribution

It establishes a new geometric characterization for solutions of anisotropic N-Laplace equations in convex cones, linking overdetermined boundary conditions to Wulff shape intersections.

Findings

Solutions exist only when the domain is a Wulff shape intersected with the cone.

The result applies under a logarithmic condition at infinity.

The proof uses a Pohozaev-type identity and anisotropic isoperimetric inequalities.

Abstract

We consider a partially overdetermined problem for anisotropic $N$-Laplace equations in a convex cone $\Sigma$ intersected with the exterior of a bounded domain $\Omega$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that $\Sigma\cap\Omega$ must be the intersection of the Wulff shape and $\Sigma$. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.