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.     

A functional limit theorem for Erdös and Rényi's law of large numbers

Summary We use the theory of large deviations on function spaces to extend Erdös and Rényi's law of large numbers. In particular, we show that with probability 1, the double-indexed set of paths {W _{N, n} } defined by where , {X _i : _i 1} is an iid sequence of random variables, andh(N)=[clogN] is relatively compact; the limit set is given by the set [xI ^*(x)1/c] whereI ^*(x) = ₀ ¹ I(x(t))dt andI is Cramér's rate function.     

Investments in stochastic maximum flow networks

Each are (i, j) of the network has capacity _ij where _ij is a non-negative random variable. The capacity of any arc may be reduced increased by an amountu _ij 0 at a cost ofc _ij u _ij . The objective is to maximizev–Kc _ij u _ij wherev is the expected maximum flow. This problem is formulated as a two-stage linear program under uncertainty. Each feasible generates a constraint where is the probability arc (i, j) is in the minimum cut set and the expected value of the maximum flow under . The formulation is later generalized to include certain conditions under which the increase in capacity of an arc may be a non-deterministic function of the investmentc _ij u _ij .     

Une propriété élementaire des fonctions dont le laplacien est constant,en particulier des fonctions harmoniques

Résumé On montre que la valeur moyenne d'une fonction harmoniqueu sur le contour d'un polygone régulier est égale à sa valeur moyenne sur les rayons du polygone (segments joignant les sommets au centre). Plus généralement si u=c, alors , oùR est le rayon du cercle inscrit.

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Summary We show that the mean value of any harmonic functionu on the boundary of a regular polygon is equal to its mean value on the radii of the polygon (segments joining all vertices to the center). More generally, if u=c, then , whereR is the radius of the inscribed circle.

     

Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scaleN

In this paper we show how wavelets originating from multiresolution analysis of scaleN give rise to certain representations of the Cuntz algebrasO _N , and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space by (S _i ) (z)=m _i (z)(z ^N ). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over . This is used to compare the usual scale-2 theory of wavelets with the scale-N theory. Also some other representations ofO _N of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.Work supproted in part by the U.S. National Science Foundation and the Norwegian Research Council.     

Fonctions Séparément Finement Surharmoniques

Nous montrons que toute fonction séparément finement surharmonique sur un ouvert de la topologie produit _{n_1}×s× _{n_k} des topologies fines des espaces R ⁿ ₁,. . ., R ⁿ _k, _{n_1}×s× _{n_k}-localement bornée inférieurement est finement surharmonique dans . On en déduit que toute fonction séparément finement harmonique, _{n_1}×s× _{n_k}-localement bornée sur est finement harmonique dans .Separately Finely Superharmonic Functions Abstract.We prove that every separately finely surperharmonic function on an open set in R ⁿ ₁×s×R ⁿ _k for the product _{n_1}×s× _{n_k} of the fine topologies on the spaces R ⁿ ₁,. . ., R ⁿ _k, _{n_1}×s× _n-_klocally lower bounded, is finely superharmonic in . We then deduce that every separateltly finely harmonic function _{n_1}×s× _{n k}-locally bounded in is finely harmonic.     

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

We study the subcritical problemsP :–u=u ^p–,u>0 on;u=0 on , being a smooth and bounded domain in ^N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P₀).

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Résumé Nous étudions les problèmes sous-critiquesP :–u=u ^p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de ^N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P₀).

     

On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems

The authors study the -limit set dichotomy of the Kolmogorov systems _i=x_i f _i(x)x _i0, 1in with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3.     

Divisibility in the K0‐Group of the Algebra of Operators on a Banach Space

Let Figiel's reflexive Banach space which is not isomorphic to its Cartesian square. We show that the K ₀group of the algebra of continuous, linear operators on contain a subgroup isomorphic to the group c ₀₀( ) of sequences rational numbers with z _n=0 eventually.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

Perturbation des résolvantes et représentation des mesures excessives

Résumé Etant donnée une résolvante V=(V )_>0 sous-markovienne sur un espace mesurable (X, B) de noyau initial V propre; on étudie alors le balayage des mesures surmédianes au moyen de la résolvante perturbée V de V par une function mesurable positive bornée sur X.Dans le cas où (X, E _v) est un espace de balayage, on montre que toute mesure excessive vérifiant , s'écrit d'une manière unique sous la forme =V où est une mesure positive sur X.     

Diophantine complexity

It is well known that any recursively enumerable set S of natural numbers can be represented by formulas of the following types a S x yx, z₁...Z_n (D (a,x,y,z₁,...,z_n)=0)¦(Davis normal form); aS z₁...z_n (E (a,z₁..., z_n)=E₂(a,z₁,...,z_n)) (exponential diophantine representation); and aS z₁... z_n, (D (a-, z₁),...)Z_n)=0) (diophantine representation). Each of these three representations yields a different measure for the complexity of S as a whole and the complexity of accepting individual members of S. A survey is presented of various results concerning such complexity measures, due to different authors.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeliniya Matematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 122–131, 1988.     

Théorème de division avec des conditions au bord

Let be an open subset of ⁿ and be a subalgebra of the algebra of analytic functions on . We suppose that satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of . We also suppose that satifies global £ojasiewicz's inequalities. We prove the following: Let andf C on flat on ; if for eacha the Taylor's serie off ata, T _a f, is in the ideal generated byT _a f ₁,...,T _a f _p in the ring of formal power series, then there exist ₁,..., _p ,C on flat on such that . This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33)     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

Generalized Quasi-Variational Inequalities Without Continuities

Given a nonempty set and two multifunctions , we consider the following generalized quasi-variational inequality problem associated with X, : Find such that . We prove several existence results in which the multifunction is not supposed to have any continuity property. Among others, we extend the results obtained in Ref. 1 for the case (x(X.     

Oscillatory Behaviour of Solutions of Two-Dimensional Differential Systems with Deviated Arguments

Sufficient conditions are established for the oscillation of proper solutions of the system

Sur un théorème de Choquet

Résumé Le but de ce travail est de caractériser les éspaces topologiques qui possèdent la propriété suivant: pour toute famille(f _t ) _{t I} de fonctions numériques définies dansX il existe une partie dénombrableI ₀ de l'ensemble d'indicesI telle que les fonctions inf _tI f _i et inf _tI0 f _i ont la meme régularisation inférieurement semi-continue.On démontre que l'espace topologiqueX possède cette propriété si et seulement si pour tout sous-ensembleA deX il existe une sous-ensemble dénombrableA ₀,A ₀ A tel queA ₀.On généralise un résultat topologique de G. Choquet.

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Global existence of solutions to one-phase Stefan problems for semilinear parabolic equations

We consider one-phase Stefan problems for the equation u _i =u _xx +u ^1+a (>0)in one-dimensional space, which have blow-up solutions for a larger initial data. In this paper, the global existence result for our problem is proved by using energy inequalities. More precisely, if >1 an initial function is sufficiently small, then the free boundary is bounded and decay in exponential order.     

Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S _n} _n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n _i for which respectively We thereby obtain conditions for to be a strong limit point of {S _n} or {S _n /n}. The first of these properties is shown to be equivalent to for some sequence a _i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n _i and a _i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.     

Fermeture en probabilité de certains sous-espaces d'un espace L 2

Summary Let _i , iI be a family of equidistributed, independant random variables, defined on a probability space (, , P). Let {f _m , mN{ be a sequence of functions such that the f _m ( _i ) are, for every i, centered random variables in L ²(, , P) and in an L _p (, , P) (where p is an even integer at least equal to 4).In this paper the closed linear subspaces of L ² (, , P) generated by the variables of the form , where for fixed M, are studied. A uniform bound of the L ^p norm of elements of the unit ball of the above defined subspaces and also the closure in probability of these subspaces is thus obtained. These results are applied to Wiener's chaos of gaussian variables.

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Equipe de Recherche n⁰ 1 « Processus stochastiques et applications » dépendant de la Section n⁰ 2 «Théories Physiques et Probabilités », associée au C.N.R.S.