One- and two-phase minimizing free boundaries in heterogeneous media

The Bernoulli free boundary problem is a classical model associated with certain interface problems from applications (capillarity, jet flows), and also arising in shape-optimization problems for Dirichlet eigenfunctions.  I will explain a result on the large-scale regularity theory of minimizing (and almost minimizing) one-phase free boundaries in periodic media, and a corresponding Liouville theorem for global minimizing solutions.  In a forthcoming work with Farhan Abedin (Lafayette College) we have also obtained analogous results in the two-phase case.  If times permits I will also discuss an application to quantitative homogenization in a shape-optimization problem for the principal Dirichlet eigenvalue.