Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

Understand

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.