We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

We investigate the diffusion limit for general conservative Boltzmann equations with oscillating coefficients. Oscillations have a frequency of the same order as the inverse of the mean free path, and the coefficients may depend on both slow and fast variables. Passing to the limit, we are led to an effective drift-diffusion equation. We also describe the diffusive behaviour when the equilibrium function has a non-vanishing flux.

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period $\epsilon$ of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates of the...

The aim of this paper is to study a class of domains whose geometry strongly depends on time namely. More precisely, we consider parabolic equations in perforated domains with rapidly pulsing (in time) periodic perforations, with a homogeneous Neumann condition on the boundary of the holes. We study the asymptotic behavior of the solutions as the period ε of the holes goes to zero. Since standard conservation laws do not hold in this model, a first difficulty is to get a priori estimates...

This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.

We study the homogenization of parabolic or hyperbolic equations like$${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}\Omega \times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in $Ø$) take possibly high values on a $\epsilon$-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

We study the homogenization of parabolic or hyperbolic equations like $${\rho }_{\epsilon }\frac{{\partial }^n{u}_{\epsilon }}{\partial t^n}-\mathrm{div}\left(a_{\epsilon }\nabla u_{\epsilon }\right)=f\text{in}Ø\times (0,T)+\text{boundary}\text{conditions},n\in \{1,2\},$$ when the coefficients ${\rho }_{\epsilon }$, $a_{\epsilon }$ (defined in Ω) take possibly high values on a ε-periodic set of grain-like inclusions of vanishing measure. Memory effects arise in the limit problem.

We investigate the asymptotic behaviour, as $\epsilon \to 0$, of a class of monotone nonlinear Neumann problems, with growth $p-1$ ($p\in ]1,+\infty [$), on a bounded multidomain ${\Omega }_{\epsilon }\subset {ℝ}^N$$(N\ge 2)$. The multidomain...

We investigate the asymptotic behaviour, as ε → 0, of a class of monotone nonlinear Neumann problems, with growth p-1 (p ∈]1, +∞[), on a bounded multidomain ${\Omega }_{\epsilon }\subset {ℝ}^N$ (N ≥ 2). The multidomain ΩE is composed of two domains. The first one is a plate which becomes asymptotically flat, with thickness hE in the xN direction, as ε → 0. The second one is a “forest" of cylinders distributed with ε-periodicity in the first N - 1 directions on the upper side of the plate. Each cylinder has a small...

We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an initial brittle bond with their neighbors. Noninterpenetration...

We give a first contribution to the homogenization of many-body structures that are exposed to large deformations and obey the noninterpenetration constraint. The many-body structures considered here resemble cord-belts like they are used to reinforce pneumatic tires. We establish and analyze an idealized model for such many-body structures in which the subbodies are assumed to be hyperelastic with a polyconvex energy density and shall exhibit an...

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies$${ℰ}_{\epsilon }\left(m\right)={\int }_{\Omega }\phi \left(x,\frac{x}{\epsilon },m\left(x\right)\right)\mathrm{d}x-{\int }_{\Omega }{h}_e\left(x\right)·m\left(x\right)\mathrm{d}x+\frac{1}{2}{\int }_{{ℝ}^3}{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x$$of a large ferromagnetic body is obtained.

A homogenization problem related to the micromagnetic energy functional is studied. In particular, the existence of the integral representation for the homogenized limit of a family of energies $${ℰ}_{\epsilon }\left(m\right)={\int }_{\Omega }\phi \left(x,\frac{x}{\epsilon },m\left(x\right)\right)\mathrm{d}x-{\int }_{\Omega }{h}_e\left(x\right)·m\left(x\right)\mathrm{d}x+\frac{1}{2}{\int }_{{ℝ}^3}{\left|\nabla u\left(x\right)\right|}^2\mathrm{d}x$$ of a large ferromagnetic body is obtained.

We consider the homogenization of both the parabolic and eigenvalue problems for a singularly perturbed convection-diffusion equation in a periodic medium. All coefficients of the equation may vary both on the macroscopic scale and on the periodic microscopic scale. Denoting by ε the period, the potential or zero-order term is scaled as ${\epsilon }^{-2}$ and the drift or first-order term is scaled as ${\epsilon }^{-1}$. Under a structural hypothesis on the first cell eigenvalue, which is assumed to admit a unique minimum in the...

Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the $\Gamma$-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.

Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the Γ-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.