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Stochastic homogenization of \(\Lambda\)-convex gradient flows. (English) Zbl 1458.49013

Summary: 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 \(\Lambda\)-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 \(p\)-Laplace operator with \(p\in(1,\infty)\). 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 \((\Lambda\)-)convex functionals.

MSC:

49J40 Variational inequalities
74Q10 Homogenization and oscillations in dynamical problems of solid mechanics
35K57 Reaction-diffusion equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
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