Stabilisation frontière d'un système d'élasticité

We prove in this Note the global existence, uniqueness and stabilization of an elasticity system with nonlinear boundary damping. This work extends results obtained by Alabau and Komornik in [1]     

Complète intégrabilité singulière

We show that an holomorphic vector field in a neighbourhood of its singular point 0 0 ∈ ℂⁿ is analytically normalizable as soon as it has a sufficiently large number of commuting holomorphic vector fields, a sufficiently large number of formal first integrals, and that a Diophantian small divisors condition related to its linear part is satisfied.     

Stabilisation frontière indirecte de systèmes faiblement couplés

This work is concerned with the boundary stabilization of coupled equations where only one of the equation is stabilized (indirect damping, see e.g. Russell [9]). We show for different examples (wave equations, Kirchhoff plates) that the full system can be strongly stabilized by that way, provided that the coupling parameter is sufficiently small (but not equal to zero). In such cases we prove that the energy decays polynomially with explicit polynomial decay rates for sufficiently smooth solutions. We extend these results to the case of two coupled wave equations with different speeds of propagation under a condition of the ratio of the two speeds and for n-dimensional intervals.     

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.     

Contre-exemple à une caractérisation conjecturée de la sphère

It has long been conjectured that a closed convex surface of class C₊² whose principal curvatures K₁, K₂ satisfy the inequality (K₁−c)(K₂−c)≤0 with some constant c, must be a sphere. Partial results have been obtained by A.D. Aleksandrov, H.F. Münzner and D. Koutroufiotis.We reformulate the conjecture in terms of hedgehogs and we give a counter-example. Besides, we prove the conjecture for surfaces of constant width and give a new proof for analytic surfaces.     

Stabilité du caractère kählérien transverse

Soient $\mathcal{F}$ un feuilletage hermitien homologiquement orientable sur une variété compacteM et $\mathcal{F}_t $ une déformation de $\mathcal{F}$ à type différentiable fixé paramétrée par un voisinageT de 0 dans ? ^d par des feuilletages transversalement holomorphes. On suppose que la métrique hermitienne transverse σ de $\mathcal{F} = \mathcal{F}_0 $ est kählérienne. On montre qu’il exist ε>0 tel que pour toutt∈T, |t|<ε le feuilletage $\mathcal{F}_t $ possède une métrique kählérienne transverseσ _t telle queσ ₀=σ; en plusσ _t dépend différentiablement det pour |t|<ε.     

Opérateur de Monge-Ampère et Tranchage des Courants Positifs Fermés

Sans résumé

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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.     

Quaternionic Monge-Ampère equations

The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ ⁿ . We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].     

Une nouvelle source d'énergie: l'énergie d'instabilité de l'atmosphère

Summary The author summarises his ideas about how to use the latent energy of a humid and unstable air mass, an integral form of solar energy. With a small addition of thermal energy, it should be possible to exert a concerted influence on meteorological phenomena. Artificial convection, either using areas of solar heating or by sudden strong combustion, should permit recovery of part of this latent energy, in the form of rain or in more concentrated forms, of which the limit-type would be ball lightning.     

Frontière de Martin Biharmonique et Représentation Intégrale des Fonctions Biharmoniques

Soit ( un espace biharmonique fort au sens de Smyrnelis dont les espaces harmoniques associés sont des espaces de Brelot qui satisfont l'axiome de proportionnalité. On définit une frontière de Martin biharmonique de qui permet de représenter les couples biharmoniques positifs, et on étudie le problème de Riquier relativement à cette frontière.Biharmonic Martin Boundary and Integral Representation of Biharmonic Functions Abstract. Let ( be a strong biharmonic space in the sense of Smyrnelis such that the associated harmonic spaces are Brelot spaces satisfying the axiom of proportionnality. We define a biharmonic Martin boundary for which permits to get the integral representation of nonegative biharmonic pairs of functions, and then we solve the Riquier problem relatively to this boundary.     

Un critère sur les petites boules dans le théorème limite central

We show that a Banach space valued random variableX such that t} \right\} = 0$$ " align="middle" border="0"> satisfies the central limit theorem if and only if the following criterion on small balls is fulfilled:

Généralisation du critère de Beurling et Nyman pour l'hypothèse de Riemann

We generalise the Beurling–Nyman criterion, already known for the Riemann ζ function, to a larger class of Dirichlet series. We link the density of some subspace of functions in $L^2(0\text{,}1)$ and the localization of the zeros of a Dirichlet series. To do so, we use the structure of the Hardy space of the half-plane. To cite this article: A. de Roton, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Caractère bien posé du problème de Cauchy pour le système de Bérenger

The Bérenger perfectly matched layer is used in computational electromagnetism as an absorbing layer in scattering problems. It raises delicate mathematical issues. In this Note we show, for regular data, the existence and uniqueness of strong solutions to the Cauchy problem derived from the PML method. The result is presented in the 2-D case. The key to the proof is an appropriate control of a mixed H¹- L² norm of the solution by the same norm of the initial data. Beside a paper is in preparation about extensions of this results (L² estimates, 3-D case) (see also [5]).     

Problèmes elliptiques à frontière libre axi-symétriques: Estimation du diamètre de la section au moyen de la capacité

Résumé Le but de cet article est d'estimer la capacité d'un pétit segment [a(1–), a(1+)]×{z=0} du démi-plan := {x = (r, z)|r > 0, z } de ² par rapport à la capacité définie par la norme . Ce résultat est ensuite utilisé pour estimer le diamètre de la surface libre de certains problèmes elliptiques non linéaires.     

Cohomologie entière de certaines variétés toriques complexes lisses de dimension trois

It is shown that the integral cohomology groups of a smooth complex 3-dimensional toric variety are free (and we compute their dimensions), under the assumption that the spherical section of its fan is homeomorphic to a closed disk. As a consequence, this gives a partial positive answer to the torsion problem in a conjecture of Reid [3] about McKay correspondence in dimension 3.     

Application de la méthode des surfaces actives et passives à l'étude des milieux hétérogènes uranium-eau légère et uranium-eau lourde

Sans résuméSubside FN 2807 du Fonds National Suisse de la Recherche Scientifique.     

Stabilisation frontière de plaques de Kirchhoff avec résolvante non compacte

Using a direct approach, we prove the asymptotic stability of Kirchhoff plates in the absence of compactness of the resolvent. We also establish the polynomial energy decay rate for the smooth solutions.