Le comportement de la résolvante modifiée du laplacien pour des obstacles captifs
Le problème de Dirichlet dans l’espace
Le problème de Pompeiu
Linearization and explicit solutions of the minimal surface equations.
We show that the apparatus of support functions, usually used in convex surfaces theory, leads to the linear equation Δh + 2h = 0 describing locally germs of minimal surfaces. Here Δ is the Laplace-Beltrami operator on the standard two-dimensional sphere. It explains the existence of the sum operator of minimal surfaces, introduced recently. In 4-dimensional space the equation Δ h + 2h = 0 becomes inequality wherever the Gauss curvature of a minimal hypersurface is nonzero.
Local uniform convergence of the Riesz means of Laplace and Dirac expansions
Locking effects in the finite element approximation of elasticity problems.
Lösung des Dirichlet Problems bei Jordangebieten mit analytischem Rand durch Interpolation.
Lower and upper bounds for the Rayleigh conductivity of a perforated plate
Lower and upper bounds for the Rayleigh conductivity of a perforation in a thick plate are usually derived from intuitive approximations and by physical reasoning. This paper addresses a mathematical justification of these approaches. As a byproduct of the rigorous handling of these issues, some improvements to previous bounds for axisymmetric holes are given as well as new estimates for tilted perforations. The main techniques are a proper use of the Dirichlet and Kelvin variational principlesin...
Majoration du nombre de résonances près de l'axe réel (d'après un travail avec M. Zworski)
Majoration en norme hölderienne de la résolvante du laplacien dans un ouvert avec coins
Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates
We are concerned with a 2D time harmonic wave propagation problem in a medium including a thin slot whose thickness ε is small with respect to the wavelength. In a previous article, we derived formally an asymptotic expansion of the solution with respect to ε using the method of matched asymptotic expansions. We also proved the existence and uniqueness of the terms of the asymptotics. In this paper, we complete the mathematical justification of our work by deriving optimal error estimates between...
Mathematical approach and numerical analysis to the fundamental solutions method of Dirichlet problem.
Mathematical modeling of time-harmonic aeroacoustics with a generalized impedance boundary condition
We study the time-harmonic acoustic scattering in a duct in presence of a flow and of a discontinuous impedance boundary condition. Unlike a continuous impedance, a discontinuous one leads to still open modeling questions, as in particular the singularity of the solution at the abrupt transition and the choice of the right unknown to formulate the scattering problem. To address these questions we propose a mathematical approach based on variational formulations set in weighted Sobolev spaces. Considering...
Mean value theorems for linear and semi-linear rotation invariant operators
Meilleures estimations asymptotiques des restes de la fonction spectrale et des valeurs propres relatifs au laplacien
Meillieurs estimations asymptotiques des restes de la fonctionn spectrale et des valeurs propres relatifs au laplacien.
Mémoire sur l'intégration des équations aux différences partielles de la Physique mathématique.
Mesures limites pour l’équation de Helmholtz dans le cas non captif
Cet article est consacré à l’étude des mesures limites associées à la solution de l’équation de Helmholtz avec un terme source se concentrant en un point. Le potentiel est supposé et l’opérateur non-captif. La solution de l’équation de Schrödinger semi-classique s’écrit alors micro-localement comme somme finie de distributions lagrangiennes. Sous une hypothèse géométrique, qui généralise l’hypothèse du viriel, on en déduit que la mesure limite existe et qu’elle vérifie des propriétés standard....
Method of interior boundaries in a mixed problem of acoustic scattering.
Méthode de calcul du coefficient de singularité pour la solution du problème de Laplace dans un domaine diédral