Méthodes d'optimisation pour l'approche de problèmes aux limites non linéaires mixtes elliptiques hyperboliques

Functional and numerical results are presented for a significative nonlinear boundary value problem of mixed type elliptic hyperbolic: a transonic flows model taking into account an entropy condition. Relation between continuous problem and its discretization in finite dimension is stated.     

Prescription du spectre du laplacien de Hodge-de Rham

We prescribe, for a Euclidean domain with given topology, any finite part of the spectrum of the Hodge and de Rham Laplacian, with multiplicity 1 or 2. The domains which yield this result do not depend on the degree of the differential forms when it is between 2 and n−1. Moreover we also prescribe the volume in this case.We get a similar result for a compact manifold. The proof relies on a generalization to differential forms of the “Cheeger's dumbbell balls” as well as on a result of convergence of the spectrum for manifolds connected together with cylinders of fixed lengths and radii which tend to 0.     

Opérateurs elliptiques sur les variétés non compactes

Soient M une variété, V un ouvert de M, et P un opérateur différentiel elliptique du second ordre, à coefficients C^∞ et réels tel que P1 0. Soit A_V l'opérateur induit par P dans l'espace de Banach C₀(V) des fonctions continues sur V nulles au point à l'infini de V, muni de la norme du suprémum. On démontre que A_V engendre un semi-groupe fortement continu à contraction ssi il existe K compact de V, h fonction continue strictement positive dans VβK et nulle au point à l'infini de V telle que (1 − P) h soit la distribution associée à une fonction continue non négative dans VβK. On en déduit immédiatement un résultat bien connu: si M est une variété de Cartan-Hadamard, A_M engendre un semi-groupe fortement continu à contraction dans C₀(M).Let M be a manifold, V an open set of M, and P an elliptic differential operator of the second order, with real C^∞ coefficients and such that P1 0. Let A_V be the operator induced by P in the complex Banach space C₀(V) of all continuous functions vanishing at the point at infinity of V, endowed with the supremum norm. One proves that A_V generates a strongly continuous contraction semi-group iff there exists K compact of V, h continuous strictly positive in VβK and 0 at infinity of V such that (1 − P) h is the distribution associated to a nonnegative continuous function in VβK. One deduces immediately from that a well-known result: if M is a Cartan-Hadamard manifold, A_M generates a strongly continuous contraction semigroup in C₀(M).     

Le bas du spectre d'une variété hyperbolique est un point selle

Besson, Courtois and Gallot proved that on a compact manifold, a hyperbolic metric has minimal entropy among the metrics of same volume. Here we show that a hyperbolic metric is a saddle point for the bottom of spectrum of the universal cover. More precisely, the bottom of its spectrum is maximal among the conformal metrics of same volume, and minimal in its Ebin class.     

Dynamique des homéomorphismes du plan au voisinage d'un point fixe

We study the dynamics of a homeomorphism of a surface near a fixed point. We compute the sequence of the Lefschetz indices of the iterates of the map. We deduce the existence of an infinite number of periodic orbits for some conservative homeomorphisms of surfaces.     

Approximation par éléments finis de problèmes elliptiques d'optimisation de forme

We consider a problem of elliptic optimal design. The control is the shape of the domain on which the Dirichlet problem for the Laplace equation is posed. In dimension n=2, S?veràk proved that there exists an optimal domain in the class of all open subsets of a given bounded open set, whose complements have a uniformly bounded number of connected components. The proof (J. Math. Pures Appl. 72 (1993) 537–551) is based on the compactness of this class of domains with respect to the complementary-Hausdorff topology and the continuous dependence of the solutions of the Dirichlet Laplacian in H¹ with respect to it. In this Note we consider a finite-element discrete version of this problem and prove that the discrete optimal domains converge in that topology towards the continuous one as the mesh-size tends to zero. The key point of the proof is that finite-element approximations of the solution of the Dirichlet Laplacian converge in H¹ whenever the polygonal domains converge in the sense of that topology. To cite this article: D. Chenais, E. Zuazua, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Une nouvelle estimation extrinsèque du spectre de l'opérateur de Dirac

We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space. To cite this article: N. Ginoux, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Une remarque sur l'annulateur du groupe des classes d'idéaux

Using an inequality for the Weil height on an Abelian extension of the rationals and a theorem of Linnik, we prove a lower bound for the index of the annihilator of the ideal class group of a cyclotomic field. This lower bound is exponential in the degree of the field. To cite this article: F. Amoroso, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

On the maximum principle for second-order elliptic operators in unbounded domainsUn principe du maximum pour opérateurs elliptiques du second ordre dans des domaines non bornés

We are concerned with the maximum principle for second-order elliptic operators of the kind Lu=a_ij(x)u_xixj+c(x)u in unbounded domains of $\text{R}{}^{\mathrm{n}}$. Using a geometric condition, already considered by Berestycki, Nirenberg and Varadhan in [2] and a weak boundary Harnack inequality due to Trudinger, Cabré [3] was able to prove the ABP (Alexandroff–Bakelman–Pucci) estimate for a large class of unbounded domains, obtaining as a consequence the maximum principle for general elliptic operators. In this Note we introduce a weak form of the above geometric condition and we show that in the case c?0 this is enough to obtain the maximum principle for a larger class of domains. To cite this article: V. Cafagna, A. Vitolo, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 359–363.     

Minorations non triviales du plus petit commun multiple de certaines suites finies d'entiers

We present here a method which allows to derive a nontrivial lower bounds for the least common multiple of some finite sequences of integers. We obtain efficient lower bounds (which in a way are optimal) for the arithmetical progressions and lower bounds less efficient (but nontrivial) for some class of quadratic sequences.In the last part of this Note, we study the integer $\mathrm{lcm}(n,n+1,\dots ,n+k)$ ($k\in \mathbb{N},n\in {\mathbb{N}}^1$). We show that it has a divisor $d_{n,k}$ simple in its dependence on n and k, and a multiple $m_{n,k}$ also simple in its dependence on n. In addition, we prove that both equalities: $\mathrm{lcm}(n,n+1,\dots ,n+k)=d_{n,k}$ and $\mathrm{lcm}(n,n+1,\dots ,n+k)=m_{n,k}$ hold for infinitely many pairs $(n,k)$. To cite this article: B. Farhi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Approximation numérique d'un problème de membrane non linéaire

We study numerically the deformations of a nonlinearly elastic membrane. We consider the nonlinear membrane model obtained by Le Dret and Raoult using Γ-convergence. In this model, membrane deformations minimize a highly nonquadratic energy. We consider a conforming finite element approximation of the problem and use a nonlinear conjugate gradient algorithm to minimize the discrete energy. To cite this article: N. Kerdid et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).