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Stochastic homogenization of $ \Lambda $-convex gradient flows

  • * Corresponding author: Mario Varga

    * Corresponding author: Mario Varga

This paper is dedicated to Alexander Mielke on the occasion of his 60th birthday

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  • 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.

    Mathematics Subject Classification: Primary: 49J40, 74Q10; Secondary: 35K57.


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