Why are the solutions to overdetermined problems usually "as symmetric as possible"?

TL;DR

This paper investigates the symmetry of nondegenerate critical points of shape functionals, showing they inherit the symmetries of the functional, with applications to overdetermined problems and a new proof regarding the uniqueness of the ball.

Contribution

It establishes that nondegenerate critical points of symmetric shape functionals inherit the same symmetries, providing new insights and a simple proof for the uniqueness of the ball in torsional rigidity maximization.

Findings

Nondegenerate critical points inherit the symmetries of the shape functional.

The ball is the unique nondegenerate critical point for the torsional rigidity maximization.

The proof avoids traditional methods like moving planes or rearrangements.

Abstract

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.