Low regularity potentials in heterogeneous Cahn--Hilliard functionals

TL;DR

This paper investigates the behavior of the Cahn-Hilliard functional with irregular potentials, establishing compactness, asymptotic analysis, and geodesic existence under weak regularity assumptions.

Contribution

It provides the first compactness and Γ-convergence results for low regularity potentials in the Cahn-Hilliard model, including geodesic existence in degenerate metrics.

Findings

Established compactness under weak hypotheses

Characterized asymptotic behavior via Γ-convergence

Proved existence of geodesics in degenerate metrics

Abstract

In this paper, we study the prototypical model of liquid-liquid phase separation, the Cahn-Hilliard functional, in a highly irregular setting. Specifically, we analyze potentials with low regularity vanishing on space-dependent wells. Under remarkably weak hypotheses, we establish a robust compactness result. Strengthening the regularity of the wells and of the growth of the potential close to the wells only slightly, we completely characterize the asymptotic behavior of the associated family of functionals through a $\Gamma$-convergence analysis. As a notable technical result, we prove the existence of geodesics for a degenerate metric and establish a uniform bound on their Euclidean length.