UBC Mathematics Department
Like the physical problems they are trying to model, many partial differential equations possess certain symmetries, i.e. they are invariant under various group actions. A natural and important question is: should the solutions inherit some or all of these symmetries? The answers to these questions are often as varied as the equations themselves and are often quite surprising and unexpected: Even scalar-valued ground states of basic energy functionals need not be symmetric even when the geometry of the domain allows them to be so. The only general principle that comes out of the literature seems to be: Symmetry properties of solutions need a proof.
In this talk, I will discuss various aspects of this question. In particular, I will explain why and how maximum principles play an important role here. I will also mention some recent results in this field, including my recent solution with Nassif Ghoussoub of the DeGiorgi conjecture in dimensions two and three.
Calculus (Math 200) is the only prerequisite for this talk.