Stochastic homogenisation for functionals defined on asymptotically piecewise rigid functions - Donnarumma, Antonio Flavio; Friedrich, Manuel (Calculus of Variations and Partial Differential Equations, 2025)
We study stochastic homogenisation of free-discontinuity surface functionals defined on piecewise rigid functions which arise in the study of fracture in brittle materials. In particular, under standard assumptions on the density, we show that there exists a Γ-limit almost surely and that it can be represented by a surface integral. In addition, the effective density can be characterised via a suitable asymptotic cell formula and is deterministic under an ergodicity assumption. We also show via Γ-convergence that the homogenised functional defined on piecewise rigid functions can be recovered from a Griffith-type model by passing to the limit of vanishing elastic deformations.
CRIS-Team Scopus Importer:2025-02-28

Γ-Convergence and Stochastic Homogenization of Second-Order Singular Perturbation Models for Phase Transitions - Donnarumma, Antonio Flavio (Journal of Nonlinear Science, 2025)
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 ε 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 Γ-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.
CRIS-Team Scopus Importer:2024-11-29
Keywords: Biomembranes; Phase transitions; Second order perturbation models; Stochastic homogenization; Γ-convergence