Stochastic homogenization of $ \Lambda $ -convex gradient flows
Martin Heida, Stefan Neukamm, Mario Varga · Discrete and Continuous Dynamical Systems - S · 2020
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a \begin{document}$ \Lambda $\end{document} -convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the \begin{document}$ p $\end{document} -Laplace operator with \begin{document}$ p\in (1, \infty) $\end{document} . The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ( \begin{document}$ \Lambda $\end{document} -)convex functionals.