Periodic unfolding and homogenizationÉclatement périodique et homogénéisation

A novel approach to periodic homogenization is proposed, based on an unfolding method, which leads to a fixed domain problem (without singularly oscillating coefficients). This method is elementary in nature and applies to cases of periodic multi-scale problems in domains with or without holes (including truss-like structures). To cite this article: D. Cioranescu et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 99–104.     

Some Remarks on Korn Inequalities

A recent joint paper with Doina Cioranescu and Julia Orlik was concerned with the homogenization of a linearized elasticity problem with inclusions and cracks(see[Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]). It required uniform estimates with respect to the homogenization parameter. A Korn inequality was used which involves unilateral terms on the boundaries where a nopenetration condition is imposed. In this paper, the author presents a general method to obtain many diverse Korn inequalities including the unilateral inequalities used in [Cioranescu, D., Damlamian, A. and Orlik, J., Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis, 82, 2013, 201–232]. A preliminary version was presented in [Damlamian, A., Some unilateral Korn inequalities with application to a contact problem with inclusions, C. R. Acad. Sci. Paris, Ser. I,350, 2012, 861–865].     

Modélisation de structures électromagnétiques périodiques – matériaux bianisotropes avec mémoire

In this Note we propose a rigorous justification of the limit constitutive law of a periodic bi-anisotropic electromagnetic structure with memory. This study is based on the periodic unfolding method, introduced by D. Cioranescu, A. Damlamian and G. Griso, and is applied on the time domain and on the frequency domain. To cite this article: A. Bossavit et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

The periodic unfolding method for a class of imperfect transmission problems

The periodic unfolding method was introduced by D. Cioranescu, A. Damlamian and G. Griso for studying the classical periodic homogenization in fixed domains and more recently extended to periodically perforated domains by D. Cioranescu, A. Damlamian, P. Donato, G. Griso, and R. Zaki. Here, the method is adapted to two-component domains which are separated by a periodic interface. The unfolding method is then applied to an elliptic problem with a jump of the solution on the interface, which is proportional to the flux and depends on a real parameter. We prove some homogenization and corrector results, which recover and complete those previously obtained by the first author and S. Monsurr`o. Bibliography: 32 titles. Illustrations: 2 figures.     

Homogenization of the 3D Maxwell system near resonances and artificial magnetism

It is now well known that the homogenization of a periodic array of parallel dielectric fibers with suitably scaled high permittivity can lead to a possibly negative frequency dependent effective permeability. However this result based on a two-dimensional micro resonator problem on the section of the fibers holds merely in the case of polarized magnetic fields, reducing thus its applications to infinite cylindrical obstacles. In this Note we propose a full 3D extension of previous asymptotic analysis based on a new averaging method for the magnetic field. We evidence a vectorial spectral problem on the periodic cell which accounts for micro-resonance effects and leads to a 3D negative effective permeability tensor. This suggests that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic split-ring structure proposed by Pendry. To cite this article: G. Bouchitté et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Interior error estimate for periodic homogenization

The aim of this Note is to give interior error estimates for problems in periodic homogenization, by using the periodic unfolding method. The interior error estimates are obtained by transposition without any supplementary hypothesis of regularity on correctors. This error is of order ?. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Estimation d'erreur et éclatement en homogénéisation périodiqueError estimate and unfolding for periodic homogenization

This Note deals with the error estimate in problems of periodic homogenization. The methods used are those of the periodic unfolding method. The error estimate is obtained without any supplementary hypothesis of regularity on correctors. To cite this article: G. Griso, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 333–336.     

Periodic homogenization for convex functionals using Mosco convergence

We study the relationship between the Mosco convergence of a sequence of convex proper lower semicontinuous functionals, defined on a reflexive Banach space, and the convergence of their subdifferentiels as maximal monotone graphs. We then apply these results together with the unfolding method (see Cioranescu et al. in C R Math Acad Sci Paris 355:99–104, 2002) to study the homogenization of equations of the form \({-\textrm{ div }d_\varepsilon=f }\), with \({(\nabla u_\varepsilon(x),d_\varepsilon(x)) \in \partial \varphi_\varepsilon(x)}\) where \({\varphi_\varepsilon (x,.)}\) is a Carathéodory convex function with suitable growth and coercivity conditions.     

Stabilisation d'un système de la thermoélasticité anisotrope avec feedbacks non linéaires

In this Note we study the stabilization of an anisotropic thermoelasticity system with (natural) Neumann boundary condition and nonlinear internal and/or boundary feedbacks. Our method consists of showing the exponential stability of the linear system by using an integral inequality, obtained by the technique of Bey et al. [E.J.D.E. 78 (2001) 1–23]; the stabilisation of the nonlinear system is deduced owing to the results from Nicaise [Rendiconti Di Matematica, Ser. VII 23 (2003) 83–116]. To cite this article: A. Heminna et al., C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Alien limit cycles near a Hamiltonian 2-saddle cycle

It is known that perturbations from a Hamiltonian 2-saddle cycle Γ can produce limit cycles that are not covered by the Abelian integral, even when it is generic. These limit cycles are called alien limit cycles. This phenomenon cannot appear in the case that Γ is a periodic orbit, a non-degenerate singularity, or a saddle loop. In this Note, we present a way to study this phenomenon in a particular unfolding of a Hamiltonian 2-saddle cycle, keeping one connection unbroken at the bifurcation. To cite this article: M. Caubergh et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Homogenization of a Ginzburg–Landau functional

We consider a nonlinear homogenization problem for a Ginzburg–Landau functional with a (positive or negative) surface energy term describing a nematic liquid crystal with inclusions. Assuming that sizes and distances between inclusions are of the same order ?, we obtain a limiting functional as $?\to 0$. We generalize the method of mesocharacteristics to show that a corresponding homogenized problem for arbitrary, periodic or non-periodic geometries is described by an anisotropic Ginzburg–Landau functional. We give computational formulas for material characteristics of an effective medium. To cite this article: L. Berlyand et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Modeling and numerical treatment of boundary data in an eddy currents problemModélisation et traitement numérique des conditions aux limites dans un problème des courants de Foucault

The aim of this paper is to analyze a finite element method to solve the eddy currents model in a bounded conductor domain. In particular we study a weak formulation in terms of the magnetic field. In order to impose suitable boundary conditions from a physical point of view, we introduce a Lagrange multiplier defined on the boundary and study the resulting mixed formulation by using classical techniques. To cite this article: A. Bermúdez et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 633–638.     

Reversible Relative Periodic Orbits

We study the bundle structure near reversible relative periodic orbits in reversible equivariant systems. In particular we show that the vector field on the bundle forms a skew product system, by which the study of bifurcation from reversible relative periodic solutions reduces to the analysis of bifurcation from reversible discrete rotating waves. We also discuss possibilities for drifts along group orbits. Our results extend those recently obtained in the equivariant context by B. Sandstede et al. (1999, J. Nonlinear Sci.9, 439-478) and C. Wulff et al. (2001, Ergodic Theory Dynam. Systems21, 605-635).     

Bandes phononiques interdites en élasticité linéarisée

In this Note we rigorously justify the existence of elastic band gaps in three-dimensional periodic composite materials with strong heterogeneities. In particular, we show how to compute these bands. To cite this article: A. Ávila et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Plate-like and shell-like inclusions with high rigidity

We study the problem of an elastic inclusion with high rigidity in a 3D domain. First we consider an inclusion with a plate-like geometry and then in the more general framework of curvilinear coordinates, an inclusion with a shell-like geometry. We compare our formal models to those obtained by Chapelle–Ferent and by Bessoud et al. To cite this article: A.-L. Bessoud et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

Existence of periodic solutions of a ordinary differential equation perturbed by a small parameter: An averaging approach

In this Note, we obtain two new results for existence of periodic solutions for differential equations perturbed by a small parameter. The first one is based on a new fixed point theorem previously obtained by the authors. The second one is based on study of suitable linearized equations. Our approach deals with degree theory and nonsmooth analysis. To cite this article: A. Gudovich et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

穿孔区域中双曲-抛物方程均匀化问题的一个注记

In this paper, we study a class of hyperbolic-parabolic problems in periodically perforated domains with a homogeneous Neumann condition on the boundary of holes. We focus on the homogenization of these equations, which generalizes those achieved by BensoussanLions-Papanicolau and Migorski. The proof is based on the periodic unfolding method in perforated domains.     

Carleman estimates and null controllability for boundary-degenerate parabolic operators

Motivated by several examples coming from physics, biology, and economics, we consider a class of parabolic operators that degenerate at the boundary of the space domain. We study null controllability by a locally distributed control. For this purpose, a specific Carleman estimate for the solutions of degenerate adjoint problems is proved. To cite this article: P. Cannarsa et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Anisotropic regularity results for Laplace and Maxwell operators in a polyhedron

As representatives of a larger class of elliptic boundary value problems of mathematical physics, we study the Dirichlet problem for the Laplace operator and the electric boundary problem for the Maxwell operator. We state regularity results in two families of weighted Sobolev spaces: A classical isotropic family, and a new anisotropic family, where the hypoellipticity along an edge of a polyhedral domain is taken into account. To cite this article: A. Buffa et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).