Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

TL;DR

This paper reviews key results on symmetry and symmetry breaking in nonlinear elliptic PDEs, emphasizing the role of flows in understanding solution rigidity and bifurcations on Euclidean spaces and manifolds.

Contribution

It introduces a unified flow-based approach to analyze symmetry, rigidity, and bifurcation phenomena in elliptic PDEs associated with variational problems.

Findings

Nonnegative solutions are unique, indicating rigidity.

Linear and nonlinear flows reveal deep properties of solutions.

Flow methods characterize symmetry breaking and bifurcations.

Abstract

The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.