Sur l'approximation uniforme des opérateurs linéaires compacts par des opérateurs non linéaires de rang fini

Sans résumé     

Spectre discret des opérateurs périodiques perturbés par un opérateur différentiel à coefficients décroissants

We study the discrete spectrum arising in the spectral gaps of an elliptic periodic differential operator, P, perturbed by a differential operator, Q, with decaying coefficient. In the semi-classical regime (Q = Q(h), h0) (resp. the large-coupling constant: Q = λQ, λ → + ∞,), we obtain an asymptotic expansion in powers of h (resp. λ) of trƒ(P + Q(h)) (resp. trƒ(P + λQ)) and we give explicitly the leading term. Here ƒ ε C₀^∞(I) where I is an interval disjoint from the spectrum of P. We apply these results to study the asymptotic distribution of eigenvalues.     

L'itération directe des opérateurs hermitiens

Sans résumé Nous dédions ce travail à Monsieur le Professeur Carathéodory à l'occasion de son soixante-dixième anniversaire.     

Un principe du maximum pour des opérateurs monotones

We prove that if T is a possibly nonlinear mapping from L¹ (φⁿ) to itself that preserves the integral, is order preserving and that commutes with translations, then T satisfies a maximum principle: the essential supremum of Tu is no greater than the essential supremum of u.     

Stabilisation par viscosité de sous-maille pour l'approximation de Galerkin des opérateurs linéaires monotones

This paper presents a stabilized Galerkin technique for approximating monotone linear operators in Hilbert spaces. The key idea consists in introducing an approximation space that is broken up into resolved and subgrid scales so that the bilinear form associated with the problem satisfies a uniform inf-sup condition with respect to this decomposition. An optimal Galerkin approximation is obtained by introducing an artificial diffusion on the subgrid scales.     

Classification projective des espaces d’opérateurs différentiels agissant sur les densités

One classifies the representation D_λμ of sl(m + 1, R) obtained by letting its projective embedding in the Lie algebra of vector fields of R^m, m > 1, act by Lie derivatives on the space of differential operators between densities of weight λ and μ. For each μ - λ, there is only finitely many isomorphism classes, most frequently one, in which case D_λμ is isomorphic to its graded space relative to the order of differentiation.     

Comparaison des semi-groupes et des résolvantes d'ordre α associés à des opérateurs différentiels de type divergence

We prove that the densities of the semi-groups of order , 0<<1 associated with differential operators of second order and of divergence type, and the density of Riesz semi-groups of order are comparables.We give a necessary and sufficient condition such that the semi-group of order and its resolvent family and their perturbated with a nonnegative and regular Radon measure are comparables.When =1, we prove that the semi-group of brownian motion and its perturbated with a radial and nonnegative measure are comparables if and only if the measure generates a bounded potential, but the result is not true if the measure is not radial.

Ce travail est soutenu par la Fondation Nationale pour la Recherche Scientifique.     

Sur les opérateurs précompacts positifs

We characterize complete Hausdorff locally convex solid lattices (E,τ) satisfying the following property: for all operators $\text{S,T}\text{:E→E}$ such that 0?S?T and T precompact, the operator S² is precompact. To cite this article: B. Aqzzouz, R. Nouira, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Un schéma linéaire vérifiant le principe du maximum pour des opérateurs de diffusion très anisotropes sur des maillages déformés

We introduce a generalized finite-difference method for anisotropic diffusion operators on distorted grids. We calculate the second-order derivatives in space using a Taylor expansion. The resulting global matrix associated to the scheme is an M-matrix. Thanks to a certain assumption on the grid properties, we show the convergence of the scheme. We show the robustness of the method in comparison with analytical solutions and results obtained by other numerical schemes. To cite this article: C. Le Potier, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Sur le problème de Cauchy pour des opérateurs faiblement hyperboliques

Sans résumé à la memoire de Lamberto Cattabriga     

Construction de solutions ramifiées autour de deux hypersurfaces caractéristiques pour des opérateurs quasi-linéaires

Let P(x,D)u=v be a partial differential equation, where P(x,D) is a quasi-linear operator of second order with simple characteristic hypersurfaces; we get solutions u which are ramified around two simple characteristic hypersurfaces if v is ramified function and D_t1v, D_t2,v are bounded.