An overdetermined problem associated to the Finsler Laplacian

TL;DR

This paper establishes a rigidity result for the anisotropic Laplacian, showing that certain overdetermined boundary value problems imply the domain is a Wulff shape, extending classical symmetry results to anisotropic settings.

Contribution

It proves that domains satisfying specific overdetermined conditions for the anisotropic Laplacian are necessarily Wulff shapes, including cases with boundary constraints on conical surfaces.

Findings

Domains are Wulff shapes under overdetermined conditions.

The result extends classical symmetry theorems to anisotropic Laplacian.

Includes cases with boundary on conical surfaces.

Abstract

We prove a rigidity result for the anisotropic Laplacian. More precisely, the domain of the problem is bounded by an unknown surface supporting a Dirichlet condition together with a Neumann-type condition which is not translation-invariant. Using a comparison argument, we show that the domain is in fact a Wulff shape. We also consider the more general case when the unknown surface is required to have its boundary on a given conical surface: in such a case, the domain of the problem is bounded by the unknown surface and by a portion of the given conical surface, which supports a homogeneous Neumann condition. We prove that the unknown surface lies on the boundary of a Wulff shape.