Homogenization in one-dimensional higher-order non-local models of phase transitions

TL;DR

This paper investigates the asymptotic behavior of higher-order fractional Cahn--Hilliard functionals with oscillations, identifying regimes and limits that differ from local models.

Contribution

It introduces a novel analysis of non-local phase transition models with oscillating coefficients, revealing new limit behaviors and scale separation effects.

Findings

Identifies three regimes based on oscillation scale and interface length.

Proves $ ext{Gamma}$-convergence to sharp-interface limits in each regime.

Highlights differences from local models due to scale separation effects.

Abstract

We study the limit behavior of Cahn--Hilliard-type functionals in which the derivative is replaced by higher-order fractional derivatives and modulated by an oscillating factor. Depending on the ratio between the oscillation scale and the interface length, we identify three different regimes and prove $\Gamma$-convergence in each regime to a suitable sharp-interface limit functional. In the extreme regimes, we prove a separation-of-scales effect that enables us to highlight the difference relative to the local models.