An exterior overdetermined problem for Finsler N-Laplacian in convex cones

Abstract

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