A partially overdetermined problem for $p$-Laplace equation in convex cones

TL;DR

This paper studies a partially overdetermined boundary value problem for the p-Laplace equation within convex cones, establishing existence, regularity, and geometric rigidity results using potential theory and inequalities.

Contribution

It introduces new existence and regularity results for capacitary potentials and proves a rigidity theorem for solutions intersecting convex cones orthogonally.

Findings

Existence and regularity of capacitary potentials in convex cones.

Rigidity result characterizing solutions under orthogonal intersection.

Application of isoperimetric and Heintze-Karcher inequalities in this context.

Abstract

We consider a partially overdetermined problem for the $p$-Laplace equation in a convex cone $\mathcal{C}$ intersected with the exterior of a smooth bounded domain $\overline{\Omega}$ in $\mathbb{R}^n$($n\geq2$). First, we establish the existence, regularity, and asymptotic behavior of a capacitary potential. Then, based on these properties of the potential, we use a $P$-function, the isoperimetric inequality, and the Heintze-Karcher type inequality in a convex cone to obtain a rigidity result under the assumption of orthogonal intersection.