The paper deals with a scalar diffusion equation $c{u}_t={\left(F\left[u_x\right]\right)}_x+f,$ where $F$ is a Prandtl-Ishlinskii operator and $c,f$ are given functions. In the diffusion or heat conduction equation the linear constitutive relation is replaced by a scalar Prandtl-Ishlinskii hysteresis spatially dependent operator. We prove existence, uniqueness and regularity of solution to the corresponding initial-boundary value problem. The problem is then homogenized by considering a sequence of equations of the above type with spatially periodic...

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 study the homogenization process of ferromagnetic multilayers in the presence of surface energies: super-exchange, also called interlayer exchange coupling, and surface anisotropy. The two main difficulties are the non-linearity of the Landau-Lifshitz equation and the absence of a good sequence of extension operators for the multilayer geometry. First, we consider the case when surface anisotropy is the dominant term, then the case when the magnitude of the super-exchange interaction is...

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

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...

The homogenization problem (i.e. the approximation of the material with periodic structure by a homogeneous one) for linear elasticity equation is studied. Both formulations in terms of displacements and in terms of stresses are considered and the results compared. The homogenized equations are derived by the multiple-scale method. Various formulae, properties of the homogenized coefficients and correctors are introduced. The convergence of displacment vector, stress tensor and local energy is proved...

In this paper we establish compactness results of multiscale and very weak multiscale type for sequences bounded in $L^2(0,T;H_0^1\left(\Omega \right))$, fulfilling a certain condition. We apply the results in the homogenization of the parabolic partial differential equation ${\epsilon }^p{\partial }_t{u}_{\epsilon }(x,t)-\nabla ·\left(a(x{\epsilon }^{-1},x{\epsilon }^{-2},t{\epsilon }^{-q},t{\epsilon }^{-r})\nabla u_{\epsilon }(x,t)\right)=f(x,t)$, where $0<p<q<r$. The homogenization result reveals two special phenomena, namely that the homogenized problem is elliptic and that the matching for which the local problem is parabolic is shifted by $p$, compared to the standard matching that gives rise to local parabolic...

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows [Probab. Theory Related Fields (2009)] and improves this latter work by...

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.