BRIANE , Marc

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies Fn, n ∈ N, defined in L2(Ω), for a bounded open subset Ω of R2. We prove that, contrary to the three dimension (or greater), the Γ-limit of any convergent subsequence of Fn is still a diffusion energy. We also provide an explicit...

In this paper we study homogenization of a linear system of elastodynamics in an elastic body with soft inclusions, which is embedded in a highly oscillating magnetic field. We show two limits behaviors according to the magnetic field. On the one hand, if the magnetic field has at least two different directions at the interface between the hard...

The paper deals with the homogenization of rigid heterogeneous plates. Assuming that the coefficients are equi-bounded in L1, we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients. This compactness result is based on a div-curl lemma for...

In this paper, a W−1,N estimate of the pressure is derived when its gradient is the divergence of a matrix-valued measure on RN , or on a regular bounded open set of RN . The proof is based partially on the Strauss inequality [Strauss, Partial Differential Equations: Proc. Symp. Pure Math. 23 (1973) 207–214] in dimension two, and on a recent...

In this paper we study homogenization of a linear system of elastodynamics in an elastic body with soft inclusions, which is embedded in a highly oscillating magnetic field. We show two limits behaviors according to the magnetic field. On the one hand, if the magnetic field has at least two different directions at the interface between the hard...

In this paper a new div-curl result is established in an open set Ω of R N , N ≥ 2, for the product of two sequences of vector-valued functions which are bounded respectively in Lp (Ω)N and Lq (Ω)N , with 1/p + 1/q = 1 + 1/(N − 1), and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product...

In this paper we prove a H-convergence type result for the homogenization of systems the coefficients of which satisfy a functional ellipticity condition and a strong equi-integrability condition. The equi-integrability assumption allows us to control the fact that the coefficients are not equi-bounded. Since the truncation principle used for...

This article deals with the links between compensated compactness and two-scale convergence. More precisely, we ask the following question: Is the div-curl compactness assumption sufficient to pass to the limit in a product of two sequences which two-scale converge with respect to the pair of variables (x, x/ε)? We reply in the negative....

This paper deals with the homogenization of nonlinear convex energies defined in W_{0}^{1,1}(\Omega )W01,1(Ω), for a regular bounded open set Ω of \mathbb{R}^{N}RN, the densities of which are not equi-bounded from above, and which satisfy the following weak coercivity condition: There exists q > N−1q>N−1 if N > 2N>2, and q⩾1q⩾1 if N = 2N=2,...

In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L1 combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two....

This paper deals with the behavior of two-dimensional linear elliptic equations with unbounded (and possibly infinite) coefficients. We prove the uniform convergence of the solutions by truncating the coefficients and using a pointwise estimate of the solutions combined with a two-dimensional capacitary estimate. We give two applications of...

The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement uεuε of an elastic body which is subjected to an oscillating magnetic field BεBε generating the Lorentz force ∂tuε×Bε∂tuε×Bε. When the magnetic field BεBε only depends on time or on space, the oscillations of BεBε induce an increase of...

Dans cet article, on obtient de nouveaux résultats de type div-rot. Pour tout ouvert Ω de RN, on étudie la limite du produit vn⋅wn où les suites vn et wn sont respectivement bornées dans Lp(Ω)N et Lq(Ω)N, alors que divvn et curlwn sont compactes dans des espaces de Sobolev, sous la condition 1⩽1p+1q⩽1+1N. L'approche utilisée repose sur une...

This paper deals with the homogenization of two-dimensional oscillating convex functionals, the densities of which are equicoercive but not uniformly bounded from above. Using a uniform-convergence result for the minimizers, which holds for this type of scalar problems in dimension two, we prove in particular that the limit energy is local and...

The present paper deals with the asymptotic behavior of equi-coercive sequences {Fn} of nonlinear functionals defined over vector-valued functions in W1,p 0 (Ω)M , where p > 1, M ≥ 1, and Ω is a bounded open set of RN , N ≥ 2. The strongly local energy density Fn(·, Du) of the functional Fn satisfies a Lipschitz condition with respect to the...