Méthodes de calcul différentiel absolu et leurs applications

Sans résumé     

Calcul des invariants de Malgrange et de Gérard-Levelt d'un système différentiel linéaire en un point singulier irrégulier

We consider a system of ordinary differential equations (Δ) Y′(x) = A(x) Y(x), where A(x) is a matrix with a pole of the order q at x = 0. For this system, B. Malgrange has introduced the quantity i(D), which can be computed with the cyclic vector method. The present paper treats the practical evaluation of i(D) and the invariants ϱ₁, ϱ₂,…, ϱ_{q − 1}, of Gérard and Levelt without using a cyclic vector. First, we shall give a survey of the algorithm given in Hilali and Wazner (Numer. Math. 50 (1987), 429–449) by which a system (Δ) is reduced to a super-irreducible form. We show that these invariants follows directly from this form.     

Homogénéisation et compacité par compensation

In a recent paper S. Fabre, J. Mossino and the author obtained a homogenization result for a class of nonlinear problems with coefficients having specific dependence upon coordinates. Their method of proof used either Aubin's lemma or compensated compactness. In this Note, our aim is to prove that the above mentioned result can be extended to a larger class of problems, where the dependence upon coordinates is replaced by a weaker condition, in terms of divergence and curl.     

Performances d'un amplificateur différentiel à transistors

Sans résumé     

Matrice de connexion d'un système aux q-différences confluant vers un système différentiel et matrices de monodromie

We built in [6] a version with elliptic coefficients of the Birkhoff connection matrix and showed its role in the classification of regular singular q-difference systems. We give here a geometrical interpretation: when q tends to 1, P tends to a locally constant matrix P such that the (finitely many) values (a)⁻¹ (b) are the monodromy matrices of the limiting differential system (assumed to be non-resonant at 0 and ∞) at the singularities on C^*.     

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.     

D-modules et ɛ-modules associés à un opérateur différentiel à caractéristiques simples

We know by the works of J. Leray that the advanced elementary solution of a differential operator P, which is supposed to be hyperbolic with simple caracteristics over an open subset U of ℝ, satisfies to a regular holonomic system of differential equations. The task of this Note is to build such a system. Moreover, the equations of this system are satisfied by any elementary solution of P, modulo holomorphic functions over a complex neighborhood of U x U, if P is supposed to be elliptic with simple characteristics.     

Densité de l'intégrale d'aire et intégrales singulières

In his 1983 paper [3], R. F. Gundy introduced a new functional related to the Littlewood-Paley theory, called thedensity of the area integral. In this paper, we prove that this functional (although highly non-linear) can be expressed as the principal value of an explicit singular integral. This result provides us with a new and precise connection between the density of the area integral and the theory of Calderón-Zygmund operators. It does not seem to be a consequence of the standard Calderón-Zygmund-Cotlar theory, because thesign of a harmonic function in the half-space fails to have, in some appropriate sense, boundary limits.      

Equations intégrales aux deux limites variables

Sans résumé à M. Enrico Bompiani pour son Jubilé scientifique     

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.     

Sur l'intégration stochastique par rapport à une martingale hilbertienne de carré intégrable

Summary We give a complete characterization of the operator-valued processes which are integrable, in the sense of [6], with respect to a fixed Hilbertvalued square integrable martingale M. This characterisation allows to complete the theory of stochastic integration with respect to Hilbert-valued martingales. In particular, we give a construction of the process M, N (predictable compensator of MN), as well as a Hilbert-valued version of the Kunita-Watanabe inequality. Finally, we deal with the distributivity of the stochastic integral X·M with respect to the martingale M: this property can be usefully applied to obtain a simple proof of a representation theorem of Gal'tchouk-Métivier.The results of this article have been announced in Comptes Rendus Note [1].