Low regularity potentials in heterogeneous Cahn--Hilliard functionals

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.