An overdetermined problem for the anisotropic capacity

TL;DR

This paper extends classical symmetry results for overdetermined Laplace problems to the anisotropic setting using the Finsler Laplacian, showing solutions are symmetric with respect to Wulff shapes in convex domains.

Contribution

It generalizes Reichel's symmetry result from Euclidean capacity to anisotropic capacity involving the Finsler Laplacian and Wulff shapes.

Findings

Established symmetry of solutions in anisotropic capacity problems.

Extended classical Euclidean results to Finsler geometry.

Demonstrated Wulff shape as the natural symmetry domain.

Abstract

We consider an overdetermined problem for the Finsler Laplacian in the exterior of a convex domain in $\mathbb{R}^N$, establishing a symmetry result for the anisotropic capacitary potential. Our result extends the one of W. Reichel [Arch. Rational Mech. Anal. 137 (1997)], where the usual Newtonian capacity is considered, giving rise to an overdetermined problem for the standard Laplace equation. Here, we replace the usual Euclidean norm of the gradient with an arbitrary norm $H$. The resulting symmetry of the solution is that of the so-called Wulff shape (a ball in the dual norm $H_0$).