Institut Élie Cartan de Lorraine

About: Institut Élie Cartan de Lorraine is a facility organization based out in Vandœuvre-lès-Nancy, France. It is known for research contribution in the topics: Boundary value problem & Stochastic differential equation. The organization has 345 authors who have published 1084 publications receiving 15512 citations. The organization is also known as: Institut Élie-Cartan de Nancy.

Holomorphic families of $[\lambda]-$primitive themes

Abstract: This article is the continuation of [B. 13-b] where we show how the isomorphism class of a $[\lambda]-$primitive theme with a given Bernstein polynomial may be characterized by a (small) finite number of complex parameters. We construct here a corresponding locally versal holomorphic deformation of $ [\lambda]-$primitive themes for each given Bernstein polynomial. Then we prove the universality of the corresponding ``canonical family'' in many cases. We also give some examples where no local universal family exists.

Asymptotics of a vanishing period : the quotient themes of a given fresco

Abstract: In this paper we introduce the word "fresco" to denote a \ $[\lambda]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the \ $[\lambda]-$primitive case to frescos having a unique Jordan-H{o}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, so to the image of a fresco in the module of asymptotic expansions. For a fixed relative de Rham cohomology class (for instance given by a smooth differential form $d-$closed and $df-$closed) each choice of a vanishing cycle in the spectral eigenspace of the monodromy for the eigenvalue \ $exp(-2i\pi.\lambda)$ \ produces a \ $[\lambda]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham class itself. So the problem to determine which theme is a quotient of a given fresco is important to deduce possible asymptotic expansions of the various vanishing period integrals associated to a given relative de Rham class when we change the choice of the vanishing cycle. In the appendix we prove a general existence result which naturally associate a fresco to any relative de Rham cohomology class of a proper holomorphic function of a complex manifold onto a disc.

New decay rates for a Cauchy thermoelastic laminated Timoshenko problem with interfacial slip under Fourier or Cattaneo laws

Abstract: The objective of the present paper is to investigate the decay of solutions for a laminated Timoshenko beam with interfacial slip in the whole space R subject to a thermal effect acting only on one component modelled by either Fourier or Cattaneo law. When the thermal effect is acting via the second or third component of the laminated Timoshenko beam (rotation angle displacement or dynamic of the slip), we obtain that both systems, Timoshenko-Fourier and Timoshenko-Cattaneo systems, satisfy the same polynomial stability estimates in the L2 -norm of the solution and its higher order derivatives with respect of the space variable. The decay rate depends on the regularity of the initial data. In addition, the presence and absence of the regularity-loss type property are determined by some relations between the parameters of systems. However, when the thermal effect is acting via the first comoponent of the system (transversal displacement), a new stability condition is introduced for both TimoshenkoFourier and Timoshenko-Cattaneo systems. This stability condition is in the form of threshold between polynomial stability and convergence to zero. To prove our results, we use the energy method in Fourier space combined with judicious choices of weight functions to build appropriate Lyapunov functionals.

Growth control of cracks under contact conditions basedon the topological derivative of the Rice’s integral

Abstract: In the present paper we propose a simple method for dealing with growth control of cracks under contact type boundary conditions on their lips. The aim is to find a mechanism for decreasing the energy release rate of cracked components, which means increasing their fracture toughness. The method consists in minimizing a shape functional defined in terms of the Rice’s integral, with respect to the nucleation of hard and/or soft inclusions, according to the information provided by the associated topological derivative. Based on Griffith’s energy criterion, this simple strategy allows for an increase in fracture toughness of the cracked component. Since the problem is non-linear, the domain decomposition technique, combined with the Steklov-Poincar´e pseudo-differential boundary operator, is used to obtain the sensitivity of the associated shape functional with respect to the nucleation of a small circular inclusion with different material property from the background. Then, the obtained topological derivatives are used to indicate the regions, where the controls should be positioned in order to solve the minimization problem we are dealing with. Finally, a numerical example is presented showing the applicability of the proposed methodology.

Etude de la géométrie optimale des zones de contrôle dans des problèmes de stabilisation

Abstract: Dans cette these, nous traitons de l'optimisation du taux de decroissance exponentielle uniforme de l'equation des ondes sur un domaine W mono ou bidimensionnel. L'amortissement se fait a l'aide d'un feedback en vitesse egal a une certaine constante k sur un sous domaine w. Ce taux de decroissance est lie a l'abscisse spectrale m de l'operateur associe au probleme et a une quantite geometrique g, introduite par Bardos, Lebeau et Rauch dans le cas bidimensionnel. On montre que l'abscisse spectrale est derivable par rapport a k a l'origine, et on etudie cette derivee J pour approximer m par le produit de k et J. Dans la premiere partie de la these, nous etudions de facon theorique les fonctionnelles J et g. Nous caracterisons les geometries optimales dans le cas d'un intervalle ou d'un carre pour des valeurs particulieres de la contrainte d'aire. Dans le cas du carre, nous concevons un algorithme de calcul exact de la quantite geometrique dans le cas ou w est un reunion de carres base sur un nouveau theoreme d'interversion de limites. La seconde partie est dediee a l'optimisation numerique des quantites J et g a l'aide de differents algorithmes genetiques. Les resultats obtenus ne sont pas intuitifs.