On a Hölder covariant version of mean dimension

Let Γ be a infinite countable group which acts naturally on ${?}^p(\Gamma )$. We introduce a modification of mean dimension which is an obstruction for ${?}^p(\Gamma )$ and ${?}^q(\Gamma )$ to be Hölder conjugates. To cite this article: A. Gournay, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Convergence rates of Neumann problems for Stokes systems

In quantitative homogenization of the Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, this paper studies the convergence rates of the velocity in $L^2$ and $H^1$ as well as those of the pressure term in $L^2$, without any smoothness assumptions on the coefficients.     

A dual finite element complex on the barycentric refinement

A simplicial mesh on an oriented two-dimensional surface gives rise to a complex $X^{?}$ of finite element spaces centered on divergence conforming Raviart–Thomas vector fields and naturally isomorphic to the simplicial cochain complex. On the barycentric refinement of such a mesh, we construct finite element spaces forming a complex $Y^{?}$, centered around curl conforming vector fields, naturally isomorphic to the simplicial chain complex on the original mesh and such that $Y^{2?i}$ is in ${\mathrm{L}}^2$ duality with $X^i$. In terms of differential forms this provides a finite element analogue of Hodge duality. To cite this article: A. Buffa, S.H. Christiansen, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On Poincaré's and J.L. Lions' lemmas

Let Ω be a bounded, connected and simply connected open subset of ${\mathbb{R}}^N$ with a Lipschitz continuous boundary. It is shown that an irrotational vector field whose components are in $H^{\mathrm{?1}}(\Omega )$ is the gradient of a function in $L^2(\Omega )$. It is also shown that this generalization of a classical lemma of Poincaré is equivalent to a well-known lemma of J.L. Lions. To cite this article: S. Kesavan, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Analyse à deux échelles d'une suite bornée de L2 sur une sous-variété du cotangent

In this Note, we investigate the generalization of two-scale Wigner measure to the case of submanifolds more general than symplectic and involutive ones for which they have been defined. We study the concentration of a bounded family in $L^2({\mathbb{R}}^d)$ on a submanifold of the cotangent space $T^1{\mathbb{R}}^d$ for which the restriction of the symplectic form to its tangent space is of constant rank. To cite this article: C. Fermanian Kammerer, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Homogenization near resonances and artificial magnetism from dielectrics

The homogenization of periodic dielectric structures in harmonic regime usually leads to an effective permittivity tensor ${?}^{\mathrm{eff}}$. It has been observed by Bouchitté and Felbacq [Waves Random Media 7 (1997) 245–256], that in the high contrast case (high conductivity fibers), this tensor depends on the angular frequency ω. In this Note, we enlight a new effect induced by microscopic resonances which leads in parallel to a possibly negative effective permeability ${\mu }^{\mathrm{eff}}(\omega )$ (although the original medium is assumed to be nonmagnetic i.e. $\mu =1$). To cite this article: G. Bouchitté, D. Felbacq, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Inverse spectral problem for singular AKNS and Schrödinger operators on [0,1]

We consider an inverse spectral problem for singular Sturm–Liouville equations on the unit interval with explicit singularity $a(a+1)/x^2$, $a\in \mathbb{N}$. This problem arises by splitting of the Schrödinger operator with radial potential acting on the unit ball of ${\mathbb{R}}^3$. Our goal is the global parametrization of potentials by spectral data noted by ${\lambda }^a$, and some norming constants noted by ${\kappa }^a$. For $a=0$ and 1, ${\lambda }^a\times {\kappa }^a$ was already known to be a global coordinate system on $L_{\mathbb{R}}^2(0,1)$. We extend this to any non-negative integer a. Similar result is obtained for singular AKNS operator. To cite this article: F. Serier, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Critical space for the parabolic-parabolic Keller–Segel model in Rd

We study the Keller–Segel system in ${\mathbb{R}}^d$ when the chemoattractant concentration is described by a parabolic equation. We prove that the critical space, with some similarity to the elliptic case, is that the initial bacteria density satisfies $n_0\in L^a({\mathbb{R}}^d)$, $a>d/2$, and that the chemoattractant concentration satisfies $\mathrm{?}{c}_0\in L^d({\mathbb{R}}^d)$. In these spaces, we prove that small initial data give rise to global solutions that vanish as the heat equation for large times and that exhibit a regularizing effect of hypercontractivity type. To cite this article: L. Corrias, B. Perthame, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Théorème de Liouville et propriété de la moyenne biharmonique restreinte

We prove, under some conditions, that a bounded Lebesgue measurable function satisfying the restricted mean value for the biharmonic functions in ${\mathbf{R}}^n$, $n\ne 2$, or in an open set of ${\mathbf{R}}^2$ with polar complement, is constant. To cite this article: M. El Kadiri, C. R. Acad. Sci. Paris, Ser. I 340 (2005).