Diffuse Interface Energies with Microscopic Heterogeneities I: Homogenization

Abstract

We analyze Allen-Cahn functionals with stationary ergodic coefficients in the regime where the length scale $\hat{I}´$ of the heterogeneities is much smaller (microscopic) than the interface width $\hat{I}\mu$ (mesoscopic). In the main result of this paper, we prove that if the ratio $\hat{I}´\hat{I}{\mu }^{-1}$ decays fast enough compared to $\hat{I}\mu$, then homogenization effects dominate, and the $\hat{I}“$-limit of the energy is the same as if the coefficients had been replaced by their homogenized values. As a byproduct of the proof, this implies that homogenization holds in the periodic setting whenever $\hat{I}´\hat{I}{\mu }^{-1}$ vanishes with $\hat{I}\mu$, no matter how slowly. In a companion paper, we prove this is sharp: if $\hat{I}´\hat{I}{\mu }^{-1}$ decays too slowly, then improbable or atypical local configurations of the medium begin to play a role, and the $\hat{I}“$-limit may be smaller than the one predicted by homogenization theory. We refer to this as the rare events regime, and we prove that it can occur in both random and almost periodic media.

v2: This version only contains results concerning the homogenization regime; the rare events regime results are removed and improved upon in a separate paper. Minor revisions, particularly filling in a gap in the treatment of the almost periodic case (Corollary 2)