Diffuse interface energies with microscopic heterogeneities

In this talk, we consider the Allen-Cahn functional with stationary and ergodic coefficients in the regime where the length scale of the heterogeneities is much smaller (microscopic) than the interface width (mesoscopic). Results by Ansini, Braides, Chiado Piat in 2003 suggest - and prove under certain restrictions on the ratio of these two scales - that the homogenization limit takes place before the sharp interface limit. In this talk, I discuss recent work that, in setting of the periodic homogenization, extends the above result beyond the mentioned restrictions, and, in the setting of stochastic homogenization, shows that a more complex behavior occurs.