  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.
