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

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.

39 pages, 1 figure

References in corpus (5)

• Surface Tension and $\Gamma$-Convergence of Van der Waals-Cahn-Hilliard Phase Transitions in Stationary Ergodic Media
• Higher-order singular perturbation models for phase transitions
• Higher-order non-local gradient theory of phase-transitions
• Singular perturbations models in phase transitions for anisotropic higher-order materials
• Gamma-Convergence of Higher-Order Phase Transition Models