Symmetry and asymmetry of minimizers of a class of noncoercive functionals

TL;DR

This paper investigates the symmetry properties of minimizers for a class of noncoercive functionals, establishing conditions for symmetry and identifying cases of symmetry breaking in two dimensions.

Contribution

It proves that minimizers are foliated Schwarz symmetric under certain conditions and demonstrates symmetry breaking phenomena in two-dimensional cases.

Findings

Minimizers exhibit foliated Schwarz symmetry in general cases.

Symmetry breaking occurs specifically in two-dimensional scenarios.

The paper extends understanding of symmetry properties for noncoercive functionals.

Abstract

In this paper we prove symmetry results for minimizers of a non coercive functional defined on the class of Sobolev functions with zero mean value. We prove that the minimizers are foliated Schwarz symmetric, i.e. they are axially symmetric with respect to an axis passing through the origin and nonincreasing in the polar angle from this axis. In the two dimensional case we show a symmetry breaking.