Published Paper

Inserted: 25 jun 2024
Last Updated: 11 jan 2025

Journal: Journal of Nonlinear Science
Volume: 35
Number: 1
Year: 2025

Abstract:

We study the effective behavior of random, heterogeneous, anisotropic, second order phase transitions energies that arise in the study of pattern formations in physical-chemical systems. Specifically, we study the asymptotic behavior, as $\epsilon$ goes to zero, of random heterogeneous anisotropic functionals in which the second order perturbation competes not only with a double well potential but also with a possibly negative contribution given by the first order term. We prove that, under suitable growth conditions and under a stationarity assumption, the functionals $\Gamma$-converge almost surely to a surface energy whose density is independent of the space variable. Furthermore, we show that the limit surface density can be described via a suitable cell formula and is deterministic when ergodicity is assumed.

Keywords: phase transitions, biomembranes, Γ-convergence, stochastic homogenization, Second order perturbation models

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