Surface Tension and $\Gamma$-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media

Abstract

We study the large scale equilibrium behavior of Van der Waals-Cahn-Hilliard phase transitions in stationary ergodic media. Specifically, we are interested in free energy functionals of the following form \begin{equation*} \mathcal{F}^ω(u) = \int_{\mathbb{R}^{d}} \left(\frac{1}{2} φ^ω(x,Du(x))^{2} + W(u(x)) \right) \, dx, \end{equation*} where $W$ is a double-well potential and ${\phi }^{\omega }(x,\cdot )$ is a stationary ergodic Finsler metric. We show that, at large scales, the random energy ${\mathcal{F}}^{\omega }$ can be approximated by the anisotropic perimeter associated with a deterministic Finsler norm $\tilde{\phi }$. To find $\tilde{\phi }$, we build on existing work of Alberti, Bellettini, and Presutti, showing, in particular, that there is a natural sub-additive quantity in this context.

[v2]: added a necessary lemma (Lemma 2) concerning invariance under a change of basis, corrected minor typos