Stochastic unfolding and homogenization
- Heida, Martin
- Neukamm, Stefan
- Varga, Mario
2010 Mathematics Subject Classification
- 49J40 74Nxx 74Qxx 74Q05
- Stochastic homogenization, unfolding, convex functionals, Allen--Cahn equation
The notion of periodic two-scale convergence and the method of periodic unfolding are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coefficients. In this paper we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients, describing random heterogeneous materials. The notion of periodic two-scale convergence has been extended in different ways to the stochastic case. In this work we introduce a stochastic unfolding method that features many similarities to periodic unfolding. In particular it allows to characterize the notion of stochastic two-scale convergence in the mean by mere convergence in an extended space. We illustrate the method on the (classical) example of stochastic homogenization of convex integral functionals, and prove a stochastic homogenization result for an non-convex evolution equation of Allen-Cahn type. Moreover, we discuss the relation of stochastic unfolding to previously introduced notions of (quenched and mean) stochastic two-scale convergence. The method introduced in this paper extends textitdiscrete stochastic unfolding, as recently introduced by the second and third author in the context of discrete-to-continuum transition.