Abstract: Free Boundaries in Random Domains from an Invariance Principle to the Homogenization of a Free Boundary Problem

Free boundary problems are models where an unknown physical field is coupled with an unknown submanifold of the underlying physical space (the "free boundary"), i.e. the temperature around a melting crystal, which interacts naturally with its solid/liquid interface. The analysis of such problems combines ideas from geometric measure theory and harmonic analysis. This talk deals with these ideas in a random setting, specifically, the analysis of a (one-phase) Hele-Shaw type model set in a domain with many (random) microscopic obstacles. The main result is that the free boundaries converge (in the macroscopic limit) to the solution of an effective, deterministic problem. This is made possible by new pointwise estimates for linear elliptic equations in perforated domains which are used to control the geometry of the interface. Many of the ideas and tools have a parallel in statistical mechanics, in fact, at the linear level, the proof leads to an invariance principle and estimates for the transition probabilities of reflected Brownian motion on perforated domains.