Speaker
Description
In many nonconvex variational free-boundary problems, regularity theory has traditionally focused on energy minimizers. Yet stable—and, more generally, finite-index—critical points arise naturally in physical models and geometric variational constructions, and are often just as relevant. A basic question is therefore: how much of the regularity theory for minimizing free boundaries survives beyond minimizers?
Such an extension is far from automatic: many arguments for minimizers rely on comparison principles or energy-decreasing deformations that are unavailable for general critical points. I will explain the significance and main difficulties of this question, and present recent progress for the free-boundary analogue of the Allen–Cahn equation in dimension three and for the one-phase Bernoulli problem in dimensions three and four.
This talk is based on joint work with X. Fernández-Real, partly in collaboration with H. Chan and A. Figalli.