Problèmes et calculs pratiques d'electricité

Monatshefte für Mathematik -     

Stabilisation frontière de problèmes de Ventcel

We study Ventcel's problems for the wave equation and the isotropic linear elastodynamic system. The boundary observability and the exact controlabillity were established in [3]. We prove the energy decay for the elastodynamic system with stationary Ventcel's conditions. We also give a boundary feedback leading to arbitrarily large energy decay rates for the elastodynamic system with evolutive Ventcel's conditions. A spectral study proves, finally, that the “natural” feedback is not sufficient for the exponential decay in the case of the wave equation with Ventcel's conditions.     

Sous-Solutions de Problèmes aux Limites en Dimension 1

We consider a boundary value problem where is a maximal monotone operator in and E is a multivalued operator in ² with non decre asing resolvent. We introduce a condition on E which insures that the operator inL ¹ (0,1) associated to this problem has non decreasing resolvent, and caracterise the subsolutions of the problem. We give different examples of operatorsE satisfying this condition.     

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].     

Problèmes d'extension dans les domaines convexes de type fini

Résumé Soient donnés D un domaine borné de , convexe, de type fini, X une hypersurface complexe telle que soit non vide, connexe et transverse et . Nous nous intéressons au problème suivant. Sous quelle(s) condition(s) existe-t-il telle que Φ|X = φ. Nous allons donner une condition nécessaire à l'existence de l'extension Φ puis sous une condition un peu plus forte nous montrerons la continuité d'un opérateur d'extension de type Berndtsson-Andersson.      

Problèmes de Skolem sur les champs algébriques

We extend Rumely's local-global principle (as refined by Cantor, Roquette, and the author, i.e. with local splitting conditions) to the case of algebraic stacks, in Artin's sense, over rings of (S-)integers of global fields. The nongeometrically connected case is also taken into account, as well as (in some instances) the case where local conditions are imposed at all places.     

Problèmes de prediction pour le processus de Wiener a deux parametres

Summary We give a decomposition of the two-parameter real Wiener process (W(x, y), (x, y)(⁺)² by using the Fourier transform defined on the group which leave it invariant. Then we use this decomposition to obtain some prediction results for this process, which allow us to give upper and lower bounds for the correlation coefficient of the vector-spaces respectively generated by the two families {W(x, y), xy s} and {W(x, y), xy t}, 0s t, {-1,1}.     

Fonctionnelles analytiques et problèmes de Cauchy

In this Note, following ideas first initiated by Fantappiè ([L. Fantappiè, Sur les méthodes nouvelles d'intégration des équations aux dérivées partielles au moyen des fonctionnelles analytiques, in : La théorie des équations aux dérivées partielles, vol. 81, Coll. Int. CNRS, Nancy, 9–15 avril 1956. [2]], [F. Pellegrino, Problèmes concrets d'analyse fonctionnelle, P. Levy (Ed.), Gauthier-Villars, Paris, 1951, pp. 357–484. [7]]), we make a link between Cauchy problems, analytic functionals and functional calculus. In particular, using classical representation formulas from complex analysis, we obtain an integral representation formula for solutions to complex Cauchy problems. Moreover, recent developments in complex convex analysis of Andersson–Passare–Sigurdsson allow us to obtain a domain of definition of the solution if Cauchy data are defined in a $\mathbb{C}$-convex domain. To cite this article: S. Rigat, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Problémes de riemann singuliers

One construct simple hyperbolic systems of conservation laws for which some Riemann problem does not have any solution in the distributionnal sense. However those systems are endowed with the best algebraic properties of the theory : they may be strictly hyperbolic with either genuinely nonlinear or linearly degenerate characteristic fields. The study of a viscous profile suggests that a solution would belong on a set of generalized functions. Lastly, one gives a related example in which a solution exists but is unbounded for any positive time     

Opérateur de Monge-Ampère et Tranchage des Courants Positifs Fermés

Sans résumé

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Isolated singularities of Monge-Ampère equations

Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s ²=E in the punctured disk 0<(x–x ₀)²+(y–y ₀)²<². Assume thatq is continuous at (x₀, y₀). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x ₀,y ₀) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344).     

拋物型Monge-Ampère方程

<正>这篇文章研究的是方程 H_r+2K_s+L_t+N(rt-s~2)=0.这是一个特殊形式的二阶非线性方程,称为Monge-Ampère方程(简写为 M-A 方程),有椭圆、双曲和抛物型之分.已有不少人研究过椭圆和双曲型的情况,这篇文章试图对抛物型 M-A 方程作-些研究.一般的二阶非线性方程     

Singular Solutions to Monge-Ampère Equation

We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side.     

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.