  In this paper, we study the symmetry properties of nondegenerate critical
points of shape functionals using the implicit function theorem. We show that,
if a shape functional is invariant with respect to some continuous group of
rotations, then its nondegenerate critical points (bounded open sets with
smooth enough boundary) share the same symmetries. We also consider the case
where the shape functional exhibits translational invariance in addition to
just rotational invariance. Finally, we study the applications of this result
to the theory of one/two-phase overdetermined problems of Serrin-type. En
passant, we give a simple proof of the fact that the ball is the only
nondegenerate critical point of the Lagrangian associated to the maximization
problem for the torsional rigidity under a volume constraint. We remark that
the proof does not rely on either the method of moving planes or rearrangement
techniques.
