A note on spatially inhomogeneous Cahn-Hilliard energies

TL;DR

This paper provides a simplified proof and generalizations for the convergence of spatially inhomogeneous Cahn-Hilliard energies to perimeter functionals, highlighting the impact of potential regularity and scale separation on the limiting behavior.

Contribution

A new, simpler proof of convergence for inhomogeneous Cahn-Hilliard energies, with extensions to stochastic and void potentials, and a modified energy avoiding strict scale separation.

Findings

Homogeneous potentials do not affect the limit.

Stronger scale separation is needed for potentials with spatially varying wells.

A modified energy functional eliminates the need for scale separation.

Abstract

In 2023, Cristoferi, Fonseca and Ganedi proved that Cahn-Hilliard type energies with spatially inhomogeneous potentials converge to the usual (isotropic and homogeneous) perimeter functional if the length-scale $\delta$ of spatial inhomogeneity in the double-well potential is small compared to the length-scale $\varepsilon$ of phase transitions. We give a simple new proof under a slightly stronger assumption on the regularity of $W$ with respect to the phase parameter. The simplicity of the proof allows us to easily find multiple generalizations in other directions, including stochastic potentials and potentials which may become zero outside the wells ('voids'). The theoretical results are complemented by numerical experiments. Our main message is that, across a wide variety of settings, nothing that looks sufficiently homogeneous at the transition length scale between phases, affects the limiting behavior in the slightest. The notable exception is the setting of potentials with spatially varying potential wells, where a stronger scale separation $\delta \ll \varepsilon^{3/2}$ is needed. Based on the analysis, we further provide a modified Modica-Mortola type energy which does not require such a scale separation. The theoretical analysis is complemented by numerical experiments.