A partially overdetermined problem for the p-Laplace equation in a convex cone

Abstract

We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $C$ intersected with the exterior of a smooth bounded domain $\overline{\Omega }$ in $R^n$ ($n\ge 2$). First, we establish the existence, regularity, and asymptotic behavior of a capacitary potential. Then, based on these properties of the potential, we obtain a rigidity result under the assumption of orthogonal intersection, by using $P$-function, the isoperimetric inequality, and a Heintze–Karcher type inequality in a convex cone.