This paper deals with homogenization of diffusion processes in a locally stationary random environment. Roughly speaking, such an environment possesses two evolution scales: both a fast microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims at giving a macroscopic approximation that takes into account the microscopic heterogeneities.

We are concerned with homogenization of stochastic differential equations (SDE) with stationary coefficients driven by Poisson random measures and Brownian motions in the critical case, that is when the limiting equation admits both a Brownian part as well as a pure jump part. We state an annealed convergence theorem. This problem is deeply connected with homogenization of integral partial diff...

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis (J. Mech. Phys. Solids, 2004) concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in t...

In this paper we obtain Meyers type (Lp+ε -) estimates for approximate solutions of nonlinear parabolic equations. This research is motivated by a numerical homogenization of these type of equations [?]. Using derived estimates we show the convergence of numerical solutions obtained from numerical homogenization methods.

In this paper, we give a review of recent results about the homogenization of particle systems with applications to Frenkel-Kontorova models and dislocation dynamics of curves. We underline the main ideas which allow us to extend the nonlinear homogenization methods for viscosity solutions, to the present framework of particles in interactions, or moving fronts in interactions. AMS Classificati...

The effects of mechanical stress during emulsification on the oxidation of methyl linoleate were investigated by four methods (high-pressure homogenization, high-speed stirring, ultrasonic homogenization, and membrane emulsification). The oxidation rates and induction periods were almost constant, regardless of the emulsification method, except for membrane emulsification, by which the inductio...

We study the qualitative homogenization of second-order Hamilton–Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient), we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends p...

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ∞ (Ω), Ω ⊂ R d) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most g...

We consider divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L ∞ (Ω), Ω ⊂ R d) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most general ...

We consider linear divergence-form scalar elliptic equations and vectorial equations for elasticity with rough (L∞(Ω), Ω ⊂ R) coefficients a(x) that, in particular, model media with non-separated scales and high contrast in material properties. While the homogenization of PDEs with periodic or ergodic coefficients and well separated scales is now well understood, we consider here the most gener...