Stochastic homogenization of \Lambda -convex gradient flows

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M2">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplace operator with <inline-formula><tex-math id="M4">\begin{document}$ p\in (1, \infty) $\end{document}</tex-math></inline-formula>. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (<inline-formula><tex-math id="M5">\begin{document}$ \Lambda $\end{document}</tex-math></inline-formula>-)convex functionals.