A Geometric approximation to the euler equations: the Vlasov–Monge–Ampère system

This paper studies the Vlasov–Monge–Ampère system (VMA), a fully non-linear version of the Vlasov–Poisson system (VP) where the (real) Monge–Ampère equation det ²/x_i x_j = substitutes for the usual Poisson equation. This system can be derived as a geometric approximation of the Euler equations of incompressible fluid mechanics in the spirit of Arnold and Ebin. Global existence of weak solutions and local existence of smooth solutions are obtained. Links between the VMA system, the VP system and the Euler equations are established through rigorous asymptotic analysis.     

Un modéle pseudo-tridimensionnel d'écoulement de fluides parfaits et généralisation de l'équation pour la fonction de courant

Résumé Dans une région des l'espace on considère l'écoulement d'un fluide et on fait l'hypothèse qu'il est pseudotridimensionnel, c'est-à-dire que son mouvement est stationnaire et qu'il existe un système de coordonnées curvilignes (y ¹,y ²,y ³) tel que les surfaces coordonnéesy ³=constante sont des surfaces de courant. Cette hypothèse entraîne l'existence d'une fonction de courant (y ¹,y ²,y ³). Si le fluide est parfait et barotrope on obtient que ³/ le long des lignes de courant oú ³ est la troisième composante contravariante du rotationnel de la vitesse et la densité de masse. Cette propriété permet de formuler une équation pour la fonction de courant, généralisant l'équation habituelle de la mécanique des fluides plans.

---

Let us consider a flow motion in a given special domain. We make the assumption that the movement of the fluid is pseudo-tridimensional, i.e., the flow motion is steady and there exists a system of curvilinear coordinates (y ¹,y ²,y ³) such that stream surfaces are defined byy ³=constant. Under this assumption there exists a stream function .Whenever the fluid is perfect and barotropic, we show that ³/= constant on stream lines (where ³ is the 3rd contravariant component of the speed's curl and is the density).Using this property we are able to write an equation for the stream function, which embeds the usual equation for two-dimensional fluid movements.

     

Problèmes de Sturm-Liouville: bornes pour les valeurs propres et les zéros des fonctions propres

Résumé Par un partage de [a, b] en deux intervalles partielsI=[a, y] etI=[y, b], on définit deux nouveaux problèmes aux valeurs propres. A partir d'une fonction particulière, on détermine, à l'aide de l'inégalité de Barta et du processus d'itération de Schwarz, des bornes inférieuresv _n ^– (y) pour la première valuer propre du problème défini dansI. On obtient de manière analogue des bornes supérieures _n ⁺ (y)pour la première valeur propre du problème défini dansI. Aun-iéme pas de l'itération, une borne inférieure pour le zéroz de la deuxième fonction propre est donnée par la racine de l'équationv _n ^– (y)— _n ⁺ (y)=0. On trouve de façon semblable, une suite de bornes supérieures tendant versz.On généralise le procédé pour les zéros des fonctions propres supérieures.

---

Summary Cutting [a, b] into two partial intervalsI=[a, y] andI=[y, b], we define two new eigenproblems. Starting from a particular function and using Barta's inequality and Schwarz's interation procedure, we determine lower boundsv _n ^– (y) for the first eigenvalue of the problem defined inI. We can likewise obtain upper bounds _n ⁺ (y) for the first eigenvalue of the problem defined inI. Aftern iterations, a lower bound for the zeroz of the second eigenfunction is the root of the equationv _n ^– (y)— _n ⁺ (y)=0. — We find, in an analogous way, a sequence of upper bounds which tend toz.We generalize this procedure for the zeros of higher eigenfunctions.

     

Global Hölder estimates for equations of Monge-Ampère type

Summary We prove interior oscillation and global Hölder estimates, independent of any boundary data, for convex solutions of certain types of Monge-Ampère equations under suitable conditions on the equation and the domain ⁿ . We also deduce the existence, uniqueness, regularity and unboundedness, under suitable conditions, of convex extremal solutions of certain Monge-Ampère equations.     

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

A higher-dimensional partial Legendre transform, and regularity of degenerate Monge-Ampère equations

In dimension n?3, we define a generalization of the classical two-dimensional partial Legendre transform, that reduces interior regularity of the generalized Monge-Ampère equation to regularity of a divergence form quasilinear system of special form. This is then used to obtain smoothness of C^2,1 solutions, having n-1 nonvanishing principal curvatures, to certain subelliptic Monge-Ampère equations in dimension n?3. A corollary is that if k?0 vanishes only at nondegenerate critical points, then a C^2,1 convex solution u is smooth if and only if the symmetric function of degree n-1 of the principal curvatures of u is positive, and moreover, u fails to be when not smooth.     

Remarks on the analyticity of the free boundary for the one-dimensional Stefan problem

Sunto Si dimostra che nel problema di Stefan (con una o due fasi) nelle variabili x(-,+) e t0il contorno libero è una funzione analitica di t ^1/2 per t0,se il dato iniziale è assunto in una classe di fungioni intere. I coefficienti dello sviluppo del contorno libero in serie di poterne di t^1/2 sono determinati mediante formule ricorrenti. Un particolare esame è svolto per il caso in cui non c'è raccordo tra il dato iniziale e quello sul contorno libero, dimostrando l'esistenza di soluzioni dotate di contorno analitico, sotto opportune limitazioni sul salto tra i due dati; sono peró messi in evidenza anche casi in cui il problema non ammette soluzioni. Infine una breve analisi è svolta per i problemi in domini limitati, fornendo una dimostrazione della analiticità del contorno libero rispetto a t per t>0,alternativa a quella di [1].

---

---

Work partially supported by the Italian CNR, while the first of the Authors was visiting the Mathematical Institute «U. Dini» of the University of Florence.     

Sur le théorème de Berry-Esseen pour les suites faiblement dépendantes

Résumé Nous étendons la méthode de démonstration du théorème de Berry-Esseen proposée par Bergström aux suites de variables aléatoires faiblement dépendantes. En particulier, nous montrons que, pour les suites stationnaires de variables aléatoires réelles bornées, la vitesse de convergence dans le théorème limite central en distance de Lévy est de l'ordre den ^–1/2 dès que la suite ( _p)p>0 des coefficients de mélange uniforme satisfait la condition _p>0 p _p <

---

About the Berry-Esseen Theorem for weakly dependent sequences

We extend the method of Bergström for the rates of convergence in the central limit theorem to weakly dependent sequences. In particular, we prove that, for stationary and uniformly mixing sequences of real-valued and bounded random variables, the rate of convergence in the central limit theorem is of the order ofn ^–1/2 as soon as the sequence ( _p)p>0 of uniform mixing coefficients satisfies _p>0 p _p <.

     

Regularity of ∫Ω|∇u|2 + λ∫Ω|u−f{2 and some gap phenomenon

We study the regularity of the minimizer u for the functional F (u,f)=|u|² + |u–f{² over all maps uH ¹(, S ²). We prove that for some suitable functions f every minimizer u is smooth in if ₀ and for the same functions f, u has singularities when is large enough.

---

Résumé On étudie la régularité des minimiseurs u du problème de minimisation min_ueH ¹(,S²)(|u|² + |u–f{². On montre que pour certaines fonctions f, u est régulière lorsque ₀ et pour les mêmes f, si est assez grand, alors u possède des singularités.

     

Vitesses maximales de décroissance des erreurs et tests optimaux associés

Summary Under reasonable conditions on , both errors of the tests that the law of a sample of size n belongs to , go to zero like exp[–n] and exp[– n]. We shall determine the best possible values for and and give a construction of sequences of tests for which the errors decrease with such an optimal rate.

---

---

Je tiens à remercier tout particulièrement G. Tusnády pour de nombreuses et très utiles suggestions et critiques concernant une première version de cet article.     

Convergence in probability for perturbed stochastic integral equations

Summary In this work, one considers two stochastic integral equations indexed by some parameter and one studies the contiguity of their solutions when the parameter converges to some ₀. Two types of behaviour are described; they lead to the notion of regular and singular perturbations. The method which is used also enables a study of the rate of convergence. Applications to time discretization of equations are given.     

Algorithmique et calculs de complexité pour un solveur de type dissections emboîtées

Résumé On considère la méthode de dissections emboîtées basée sur des théorèmes de séparation introduite par Gilbert-Tarjan et Roman utilisée pour la résolution par élimination de Gauss de grands systèmes linéaires creux. Plus précisemment, on étudie une structure de données par blocs similaire à celle proposée par George dans le cadre des graphes en grille, et on démontre les propriétés suivantes: d'une part, pour des familles de graphes à degré borné admettant unn -théorème de séparation, 1/2<1, le stockage secondaire de la structure par blocs contenant la matrice factorisée est linéaire par rapport à la taille du système; d'autre part, en rajoutant une hypothèse non restrictive sur la manière d'effectuer la séparation, la structure peut être construite en temps linéaire par une factorisation logique par blocs. Des exemples numériques illustrent ces résultats théoriques.

---

Algorithmic study and complexity bounds for a nested dissection solver

Summary We consider the nested dissection method based on separator theorems introduced by Gilbert-Tarjan and Roman used for solving large sparse systems of linear equations. More precisely, we study a block storage scheme such as proposed by George for regular square grids and we prove the following results: first, for families of graphs of bounded degree withn -separator theorem, 1/2<1, the overhead storage of the block data structure for the factored matrix is linear in system dimension; on the other hand, by adding a non restrictive assumption on the separation, this structure can be constructed by a block symbolic factorization which runs in time linear in matrix dimension. Numerical experiments illustrating these theoretical results are provided.

     

On the minimizers of the relaxed energy functional of mappings from higher dimensional balls into S 2

We prove a new approximation theorem, which enables us to show that the relaxed energy of Sobolev mappings u from higher dimensional balls into S² is given by , provided their singular set is of Lebesgue measure zero. Here is the mass of the minimal integer multiplicity connection associated to the singularity current S_u of u. Using this approximation theorem, we prove a partial regularity theorem for minimizers of the relaxed energy functional.Received: 5 May 2004, Accepted: 19 October 2004, Published online: 10 December 2004     

Membranes élastiquement liées inhomogènes ou sur une surface: Une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn

Résumé Soit ₁ la première valeur propre d'une membrane élastiquement liée inhomogène ou sur une surface. On caractérise ₁ comme Maximum, avec une méthode inspirée par celle des longueurs extrémales de Beurling et Ahlfors [1]. Cette caractérisation permet une nouvelle extension du théorème isopérimétrique de Rayleigh-Faber-Krahn ([10], p. 191 ou [4], p. 104).

---

Summary Let ₁ be the first eigenvalue of an elastically supported membrane, either inhomogeneous or on a surface. ₁ is caracterized as a Maximum by a method close to the extremal length of Beurling and Ahlfors [1]. This caracterization allows a new extension of Rayleigh-Faber-Krahn's isoperimetric theorem ([10], p. 191 or [4], p. 104).

---

---

Cette recherche a été soutenue par le Fonds national suisse de la recherche scientifique.     

On computing the number of subgroups of a finite Abelian group

We consider the numberN _A (r) of subgroups of orderp ^r ofA, whereA is a finite Abelianp-group of type =₁,₂,..., _l ()), i.e. the direct sum of cyclic groups of order ⁱⁱ. Formulas for computingN _A (r) are well known. Here we derive a recurrence relation forN _A (r), which enables us to prove a conjecture of P. E. Dyubyuk about congruences betweenN _A (r) and the Gaussian binomial coefficient .     

n-th roots of a non-negative operator. Conditions for uniqueness

In the present paper we study which restrictions must be imposed on the n-th roots of certain non-negative closed operators A on a Banach space so that these roots are unique.Counterexamples are given to show that the two results on this subject in the previous literature are incorrect.Finally, we obtain an explicit formula relating the canonical root A^1/n to another given non-negative n-th root B, and this allows us to establish the conditions for a given element to yield the same value by both roots. This point of view, which had not been considered up to now, provides simple conditions for global uniqueness that need only to be checked on the subspaces D(A)=_n1D(Aⁿ) and R(A)=_n1R(Aⁿ).     

Construction d'une paramétrixe pour des opérateurs différentiels hypoelliptiques marimaux sur un espace homogéne d'un groupe de Lie nilpotent gradué de rang 2

Soient G une alébre de Lie nilpotente stratifée de rang 2, une sous-algébre de G, _0, la représentation de G dans l'espace L ²( \ G) indiute par le caractére trivial C, P un opérateur homogène appartenant à l'algébre universelle enveloppante (complexifiée) U(G) tel que l'opérateur _0, (P) soit hypoelliptique maximal. Cet opérateur peut s'exprimer par une intégrale dépendant de la restriction du symbole p de P au sousensemble = G · décrit par les orbites des éléments de dans la représentation contragrédiente de G dans G ^*.Une algèbre de symboles définis sur est construite et permet de déterminer une paramétrixe de _0, (P); des résultats de réguralité de cet opérateur dans des espaces de Sobolev adaptés sont ensuite obtenus.     

Backward doubly stochastic differential equations and systems of quasilinear SPDEs

Summary We introduce a new class of backward stochastic differential equations, which allows us to produce a probabilistic representation of certain quasilinear stochastic partial differential equations, thus extending the Feynman-Kac formula for linear SPDE's.The research of this author was partially supported by DRET under contract 901636/A000/DRET/DS/SRThe research of this author was supported by a grant from the French Ministère de la Recherche et de la Technologie, which is gratefully acknowledged     

A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in W -1/ q, q

We develop a theory for a general class of very weak solutions to stationary Stokes and Navier-Stokes equations in a bounded domain with boundary of class C^2,1, corresponding to boundary data in the distribution space W^–1/q,^q(), 1<q<. These solutions exist and are unique (for small data, in the nonlinear case) in their class of existence, and satisfy a correponding estimate in terms of the data. Moreover, they become regular if the data are regular. To our knowledge, the only existence result for solutions attaining such boundary data is due to Giga, [16], Proposition 2.2, for the Stokes case. However, the methods and the approach used in the present paper are different than Gigas and cover more general issues, including the nonlinear Navier-Stokes equations and the precise way in which the boundary data are attained by the solutions. We also introduce, in the last section, a further generalization of the solution class.Acknowledgement We would like to thank Professors Herbert Amann and Remigio Russo for helpful conversations on aspects of this research. In particular, Amanns recent paper [4] for the nonstationary case was for us a motivation to study the steady case. We also would like to thank an anonymous referee whose comments helped the clarity of the formulation of the trace theorem. This paper was completed while G.P.Galdi was a Deutsche Forschungsgemeinschaft (DFG) Mercator Professor at the University of Paderborn in the period May-August 2003.     

Sur les équations fonctionnelles auxq-différences

Summary In this paper, we study the convergence of formal power series solutions of functional equations of the formP _k(x)(^[k](x))=(x), where ^[k] (x) denotes thek-th iterate of the function.We obtain results similar to the results of Malgrange and Ramis for formal solutions of differential equations: if(0) = 0, and(0) =q is a nonzero complex number with absolute value less than one then, if(x)=a(n)x ⁿ is a divergent solution, there exists a positive real numbers such that the power seriesa(n)q ^sn(n+1)2 x ⁿ has a finite and nonzero radius of convergence.