Isoperimetric cones and minimal solutions of partial overdetermined problems

TL;DR

This paper characterizes the minimal solutions of a partial overdetermined boundary value problem in cones, showing that spherical sectors centered at the cone's vertex are uniquely optimal in isoperimetric cones, and extends isoperimetric properties to cones close to these special cones.

Contribution

It proves that in isoperimetric cones, the only solutions minimizing torsional energy are spherical sectors, and generalizes isoperimetric properties to cones near these special cones.

Findings

Spherical sectors are the unique minimizers in isoperimetric cones.

Cones close to isoperimetric cones are also isoperimetric.

Improved characterization of constant mean curvature graphs in cones.

Abstract

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that in cones having an isoperimetric property the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].