Existence of nonradial domains for overdetermined and isoperimetric problems 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.

Overdetermined problems and relative Cheeger sets in unbounded domains

“…For nonconvex cones a similar characterization has been proved, but only for CMC radial graphs ( [15,16]). On the other side, nonradiality results have been proved for a class of nonconvex cones ( [7]).…”