An overdetermined problem related to the Finsler -Laplacian

Abstract

In this paper, we consider the Finsler $$-Laplacian torsion equation. The domain of the problem is bounded by a conical surface supporting a Neumann-type condition, and an unknown surface supporting both a Dirichlet and a Neumann condition. The case when the cone coincides with the punctured space is included. We show that the existence of a weak solution implies that the unknown surface lies on the boundary of a Finsler-ball. Incidentally, some properties of the Finsler–Minkowski norms are proved here under mild smoothness assumptions.