$\Gamma$-convergence and stochastic homogenization of second order singular perturbation models for phase transitions

Antonio Flavio Donnarumma

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arXiv · 2406.14356

$\Gamma$-convergence and stochastic homogenization of second order singular perturbation models for phase transitions

Antonio Flavio Donnarumma

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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.

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Antonio Flavio Donnarumma. 2024-06-20. $\Gamma$-convergence and stochastic homogenization of second order singular perturbation models for phase transitions. https://arxiv.org/abs/2406.14356

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