arXiv subjects

Luca Pignatelli

Publications and source records attributed to Luca Pignatelli.

2 recordsLinked to original sources

arXiv2026

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.

math.AP↗

arXiv2025

Phase separation in multiply periodic materials with fine microstructures

We study a Cahn-Hilliard model for phase separation in composite materials with multiple periodic microstructures. These are modeled by considering a highly oscillating potential. The focus of this paper is in the case where the scales of the microstructures are smaller than that of phase separation. We provide a compactness result and prove that the {\Gamma}-limit of the energy is a multiple of the perimeter. In particular, using the recently introduced unfolding operator for multiple scales, we show that the taking the limit of all of the scales together is equivalent to taking one limit at the time, starting from the smaller scale and keeping the larger fixed.

math.AP↗