Observabilité, contrôlabilité et stabilisation frontière du système d’élasticité linéaire

We study homogeneous but not necessarily isotropic linear elastodynamic systems. Applying the Hilbert Uniqueness Method (HUM), we establish their boundary observability and their exact boundary controllability. Then we construct boundary feedbacks leading to arbitrarily large energy decay rates. Finally, under some geometrical conditions, we prove that a “natural” boundary feedback leads also to exponentiel energy decay.     

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).     

Rationalité des séries de poincaré et des fonctions zêta motiviques

Soient k un corps de caractéristique 0 et X une k[[t]]-variété (éventuellement singulière) plate, purement de dimension relative d. Nous prouvons la rationalité des séries de Poincaré motiviques et de fonctions Zêta dIgusa motiviques, associées à X, à laide de lintégration motivique, du théorème de désingularisation plongée dHironaka, de la théorie des modèles de Néron faibles pour les schémas formels et dun théorème délimination des quantificateurs en théorie des modèles.Revised version: 17 March 2004     

Axiome de proportionnalité et représentation intégrale des potentiels

On montre que dans un espace -harmonique X à base dénombrable, satisfaisant l'axiome de proportionnalité et admettant un noyau de Green G, tout potentiel se représente par G au moyen d'une mesure de Radon positive unique et on en déduit que X vérifie la propriété de convergence de Doob. Mathematics Subject Classification (2000): 31D05.     

Dérivabilité globale et locale des fonctionnelles de Wiener

In this Note we give sufficient conditions which imply the global Sobolev differentiability of locally differentiable Wiener functionals.     

Comportement asymptotique des solutions du système de l'élasticité linéarisée anisotrope hétérogène dans des cylindres minces

We study the convergence of the solution u^ɛ of an anisotropic, heterogeneous, linearized elasticity problem in a cylinder, the diameter of which tends to zero. We prove in particular that u^ɛ − (u + ɛv + ɛ²w) strongly converges to zero (in a sense which will be specified), where (u, v, w) is the unique solution of an elliptic system of partial differential equations.     

Sur certains problèmes et théorèmes concernant la continuité et la dérivabilité des fonctions

Sans résumé     

Homogénéisation de frontière en élasticité linéaire

We describe the asymptotic behaviour of the solution of a linear elastic problem posed in a domain of R³, with homogeneous Dirichlet boundary conditions imposed on small zones of size less than ɛ distributed on some part of the boundary of this domain, when the parameter ɛ tends to 0. We use epi-convergence arguments in order to establish this asymptotic behaviour.     

Comportement post-critique des poutres élastoplastiques: existence et régularité des branches bifurquées

In a previous Note dealing with bifurcated solutions of the quasi-static equilibrium problem of an elastic-plastic compressed beam, we established that the loading axis contains, on the fundamental branch, successive intervals completely filled with points where the velocity answer is not unique. This Note shows that there exists actually a bifurcated branch tangent to any of these non trivial velocities. The branches are the solutions of Cauchy problems whose initial data are the corresponding points of the fundamental branch. From these problems, we deduce the monotonicity of the branches, and their regularity, depending on the regularity of the cross-section.     

Observabilité et contrôlabilité exacte indirecte interne par un contrôle localement distribué de systèmes d'équations couplées

We study the observability and the exact controllability of a weakly coupled system with an internal locally control acting on only one equation. We show that, for sufficiently large time, the observation of the velocity of the first component of the solution on a neighborhood of a part of the boundary allows us to get back a weakened energy of initial data of the second component, this if the coupling parameter is sufficiently small, but non-vanishing. This result leads to a uniqueness theorem and, by the HUM method, we prove that the total system is exactly controllable.     

Sur l'existence et l'unicité des solutions pour des équations de Drude–Born–Fedorov homogénéisées en domaine borné et applications aux métamatériaux

A general Drude–Born–Fedorov (DBF) system, with materials like metamaterials, on a bounded domain is studied. The well-posedness of this kind of systems cannot be investigated with usual results for the DBF system establishing electromagnetic phenomenons in chiral materials. An existence and uniqueness result for these generalized DBF systems is then shown under compatible assumptions relevant for some models from literature. At last, homogenized dielectric–chiral photonic crystals and convex Perfectly Mached Layers (“PML”) are studied.     

Les coefficients cinematiques d’élasticité

Annali di Matematica Pura ed Applicata (1923 -) - L’auteur propose de modifier la loi deHooke de façon à exprimer le travail élastique en function de l’unité de...     

Annulation et pureté des groupes de cohomologie rigide associés à des sommes exponentielles

We show a purity property for the rigid cohomology groups associated to exponential sums on a smooth and affine variety. For this purpose, we compare these groups with the p-adic Banach spaces of Dwork's theory constructed by Adolphson and Sperber [1].     

Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire

We present a simple proof of regularity and compactness phenomena induced by singular Boltzmann collision kernels.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Connexité des sous-niveaux des fonctionnelles intégrales

Let (T, ℐ, μ) be a σ-finite atomless measure space,p∈[1,∞),E a real Banach space andf a measurable function:E xT→ℝ. We denote byF the functionalF: and byDom(F) its domain, it is the set {uεL ^p(T,E):ū(t)=f(u),t)εL ¹(T)}, and we prove that the sublevelsS(λ)={u:F(u)≤λ} are all connected in the subspaceDom(F) of the Banach spaceL ^p(T, E).     

Normalité asymptotique de l'estimateur par polynômes locaux de la densité et ses dérivées pour des processus réels et mélangeants

We establish the convergence in probability and the asymptotic normality of the kernel polynomial estimators of the density function and its derivatives when the observed process is strictly stationary and strong mixing.