Une méthode de point fixe pour le p-laplacien

Due to the compactness of the operator (-Δ_p)¹ N_ƒ, 1 < p < ∞, where Δ_p is the p-Laplacian and N_ƒ is the Nemytskii operator corresponding to a Caratheodory function ƒ: Ω × R → R, which satisfies a particular growth condition, the homotopy invariance of Leray-Schauder degree can be used in order to prove the existence of a W₀^1,p (Ω)-solution for the equation -Δ_p u = N_ƒ u.     

Une variante à la Coxeter du groupe de Steinberg

LetG _n ()be the semi-direct product of the symmetric groupS _n by the Steinberg groupSt _n ()of a ringWe first prove thatG _n ()has a Coxeter-type presentation. The canonical morphism St _n () GL _n ()extends to a group homo G_n() GL _n ()We next determine the kernel of for n = We also give an expression for the generator of the algebraic K group K ₂(Z)of the integers in terms of permutation matrices.     

L'influence de la géométrie de Bolyai-Lobatchevsky sur le développement de la méthode axiomatique

Sans résumé     

Une majoration de la multiplicité spectrale d'opérateurs associés à des cocycles réguliers

We study the spectral multiplicity of unitary operators of defined by cocycles over an irrational rotation α. We prove that the multiplicity is finite whenever the cocycle has bounded variation and we give explicit bounds. For a cocycle given by an absolutely continuous function ϕ on [0,1], we show that the multiplicity is strictly less than max (2.•∫ϕ(x)dx•+1), which is optimal in the case ϕ(x)=nx (where the multiplicity is exactlyn). The proofs are based on the representation of the rotation as a “local rank one” transformation, which arises from the continued fraction expansion of α.      

Sur la méthode d'intégration de M. Tchébychef

Sans résumé     

Une méthode de Γ-convergence pour un modèle de membrane non linéaire

We consider a cylindrical three-dimensional nonlinear hvperelastic body whose stored energy goes to infinity when the jacobian of the deformation gradient goes to zero. We show, under appropriate hypotheses on the applied loads, that the three-dimensional energies Γ-converge to the energy of a nonlinear membrane when the thickness of the cylinder goes to zero. The Γ-convergence takes place in the weak topology of a Sobolev space. As a consequence, we show that the sequence of minimizers also converges toward a minimizer of the limit energy.     

Une méthode géométrique élémentaire pour l’étude de certaines questions de la théorie des courbes planes

Sans résumé Une partie du présent travail est la reproduction d’un article paru dans Arkiv f?r mathematik, astronomi och fysik (utg. af K. Sv. Vet.-Akademien, Stockholm), Bd. 1, p. 681.     

Nouvelle méthode d'édifier la géométrie plane de Bolyai et de Lobatchefski

Sans résumé     

Une méthode numérique décentrée pour la résolution d'écoulements diphasiques

This Note focuses on the numerical approximation of two-fluid flow models described by six balance equations. We introduce an original splitting technique especially derived to use the approximate Riemann solvers of the usual gas dynamics and to allow for a straightforward extension to various and detailed exchange source terms. When based on suitable kinetic upwind schemes, the whole scheme preserves the positivity of all the thermodynamic variables under a fairly unrestrictive “CFL like” condition. Several stiff numerical teats, are presented including phase separation, in order to highlight the efficiency of the proposed method.     

Une méthode de volumes finis pour les équations elliptiques du second ordre

We present a new finite volume method for approximating second order elliptic equations. This method has several advantages: — it allows large mesh distortions which occur in Lagrangian hydrodynamics calculations; — it is convenient for the approximation of crossed derivative terms as those which take place in magnetohydrodynamics will Hall effect; — it generalizes the finite difference method and the finite volume method using Delaunay-Voronoi meshes.     

Analyse numérique de la méthode des variables adiabatiques pour l'approximation de l'hamiltonien nucléaire

Many problems in quantum chemistry deal with the computation of fundamental or excited states of molecules and lead to the resolution of eigenvalue problems. One of the major difficulties in these computations lies in the very large dimension of the systems to be solved. Indeed, these eigenfunctions depend on 3n variables, where n stands for the number of particles (electrons or protons) in the molecule. In order to diminish the size of the systems to be solved, the chemists have proposed many interesting ideas. Among those stands the adiabatic variable method; we present in this Note a mathematical analysis of this approximation and propose, in particular, an a posteriori estimate that might allow for verifying the adiabaticity hypothesis that is done on some variables.     

Une méthode de raffinement de maillage espace-temps pour le système de Maxwell en dimension un

We propose a new method for space-time mesh refinement in the finite difference approximation of the 1D Maxwell's system. Our strategy is based on the conservation of a discrete energy.     

Une méthode des sur et sous solutions améliorée

The usual upper and lower solutions method for solving a second order nonlinear elliptic equation is iterative, with the drawback of a tricky, if not sometimes impossible, derivation of the (quite essential) uniform $C^2$ estimate, due to the occurence in the same equation of two successive iteration unknowns. We present here a method free of such a drawback, based on an elementary fixed point argument. To cite this article: P. Delanoë, C. R. Acad. Sci. Paris, Ser. I 341 (2005).