Peter Morfe

University of Chicago

Talk Title: Stochastic homogenization of Modica-Mortola type functionals

Abstract: We will review what is known about functionals of Modica-Mortola type, in particular their relation to perimeter and related functionals on BV spaces, and the corresponding results for the associated gradient flows, which include the Allen-Cahn equation. These functionals have the interpretation of free energy in the van der Waals-Cahn-Hilliard theory of phase transitions and are therefore of some interest in statistical mechanics, and a very complete rigorous description of the macroscopic behavior of such systems is available. Unfortunately, much less is known when one strays from the spatially homogeneous case and includes the effects of small-scale heterogeneities. We will sketch a recent proof of the (Gamma-) limit of Modica-Mortola functionals in general stationary ergodic media.