Existence of nonradial domains for overdetermined and isoperimetric problems in nonconvex cones

TL;DR

This paper proves the existence of nonradial domains within nonconvex cones that admit solutions to overdetermined boundary problems, using variational methods to identify conditions where spherical sectors are not minimizers.

Contribution

It introduces a variational approach to establish the existence of nonradial solutions in nonconvex cones for overdetermined and isoperimetric problems, highlighting conditions that exclude spherical sectors as minimizers.

Findings

Existence of nonradial minimizers under volume constraints.

Conditions on domain $D$ prevent spherical sectors from being minimizers.

Results extend to the relative isoperimetric problem in nonconvex cones.

Abstract

In this work we address the question of the existence of nonradial domains inside a nonconvex cone for which a mixed boundary overdetermined problem admits a solution. Our approach is variational, and consists in proving the existence of nonradial minimizers, under a volume constraint, of the associated torsional energy functional. In particular we give a condition on the domain $D$ on the sphere spanning the cone which ensures that the spherical sector is not a minimizer. Similar results are obtained for the relative isoperimetric problem in nonconvex cones.