Asymptotic expansions obtained by a center manifold theorem

Sunto Viene presentato un nuovo metodo per la determinazione degli sviluppi asintotici della soluzione esterna di sistemi di equazioni differenziali ordinarie singolarmente perturbati. Il metodo proposto, basato sulla teoria geometrica delle perturbazioni singolari e in particolare su un teorema di esistenza di varietà centrale, permette di ottenere le equazioni differenziali che definiscono le variabili « lente » senza la preventiva conoscenza dei corrispondenti sviluppi per le variabili « veloci ». Inoltre, se i sistemi vengono dati con condizioni iniziali, alcune formule che esprimono le corrette condizioni iniziali da assegnare alle equazioni differenziali trovate — formule già note nel « caso stabile » — vengono estese al « caso condizionalmente stabile »; il procedimento qui usato risulta anche più sintetico rispetto a quelli precedentemente proposti. Infine viene studiata un'applicazione ad una classe assai generale di equazioni derivanti dalla cinetica delle reazioni enzimatiche.

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Lavoro eseguito nell'ambito dei programmi del gruppo di ricerca « Equazioni di Evoluzione e Applicazioni », M.P.I., e del Gruppo Nazionale Fisica-Matematica del C.N.R.     

On a unifying approach to decomposition theorems of Yosida-Hewitt type

In this paper we deal with a very general form of the Yosida-Hewitt theorem on the decomposition of measures into countably additive («normal») and purely finitely additive («antinormal») parts. It expands a previous one by the authors with the aim of joining two different standpoints to the Yosida-Hewitt type theorems. The first goes back to the original publication defining the «antinormal» part as a certain disjoint complement to the «normal» one. The second approach goes deeper and characterizes this disjoint complement intrinsically i.e. as a measure, functional or operator which is equal to zero on a huge set. These two points of view are common for the publications connected, respectively, with measure theory and, theory of vector lattices; the second allows important applications. The unification of these approaches gives an opportunity to derive new information in the case of vector measures. We have taken the opportunity of this paper also to furnish a survey of the topic.     

A boundary-layer in an elastico-viscous fluid

Zusammenfassung Die Prandtlsche Grenzschichttheorie wird verallgemeinert, so dass sie die Nollschen «einfachen Flüssigkeiten» erfasst. Es wird bewiesen, dass bei Flüssigkeiten mit geringem «Erinnerungsvermögen» die in der Grenzschicht vorhandenen Spannungen denjenigen ähnlich sind, welche bei einer einfachen Schubbeanspruchung vorkommen.     

Wiener criterion and potential estimates for obstacle problems relative to degenerate elliptic operators

Summary We give a Wiener's type criterion for the continuity of the local solutions of obstacle problems relative to a degenerate elliptic operator. Moreover, we give an estimate on the modulus of continuity of the solutions and we also estimate the «energy decay» at a point.

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Sunto Si dà un criterio di Wiener per la continuità in un punto delle soluzioni locali di problemi d'ostacolo relativi ad un operatore ellittico degenere. Si ottiene inoltre una stima del modulo di continuità della soluzione e del «decay» dell'«energia» in un punto.

     

Quelques applications de la théorie générale des processus. I

Sans résuméEquipe de Recherche n^o 1 «Processus stochastiques et application» dépendant de la Section n^o 2 «Théories Physiques et Probabilités» associée au C.N.R.S.     

Some variations on the notion of connection

Distributions on manifolds are studied in terms of jets of submanifolds and are interpreted as «pre-connections» or «almost-fibrings»; the associated differential calculus is developed in detail. A comparison with connections on fibred manifolds is analysed. Moreover, «higher order pre-connections», defined as pre-connections dependent on jets of arbitrary order, are introduced and studied. It is shown that infinite jets play an essential role in the associated differential calculus.This work has been performed the visits of Prof. A. M.Vinogradov at the Department of Applied Mathematics, supported by Gruppo Nazionale per la Fisica Matematica of CNR (1989, 1990).This work has been partially supported by funds (40% and 60%) of MURST.     

On the optimal control and relaxation of nonlinear infinite dimensional systems

Summary In this paper we study optimal control problems for infinite dimensional systems governed by a semilinear evolution equation. First under appropriate convexity and growth conditions, we establish the existence of optimal pairs. Then we drop the convexity hypothesis and we pass to a larger system known as the « relaxed system ». We show that this system has a solution and the value of the relaxed optimization problem is equal to the value of the original one. Next we restrict our attention to linear systems and establish two « bang-bang » type theorems. Finally we present some examples from systems governed by partial differential equations.Research supported by N.S.F. Grant-8602313.Work done while on leave at the « University of Thessaloniki, School of Technology, Mathematics Division, Thessaloniki 54006, Greece ».     

Kontinua zweiter Ordnung

Ohne ZusammenfassungDie Arbeit ist eine Fortsetzung meiner Untersuchungen zu den Grundlagen der Topologie: «Stetige Mengen», Monatsh. f. Math. Phys. 31 (1921) und «Bereiche zweiter Ordnung», ebenda Bd. 32 (1922). Sie seien im Folgenden mit I und II zitiert. Vgl. auch Hausdorff, Grundzüge der Mengenlehre. 1914, S. 290–304.     

Asymptotic spherical symmetry of the free boundary in degenerate diffusion equations

Summary A general class of nonlinear degenerate parabolic equations in many space dimensions is considered and two main results concerning the free boundary are proved: (i) the «eventual» Lipschitz continuity in the space variable, (ii) the asymptotic spherical symmetry in a stronger sense than the «almost radiality» proved by Aronson & Caffarelli [2] for the porous medium equation. The proofs make use of geometric ideas based on the comparison principle and the method of moving planes.     

Cohesive categories and manifolds

Sunto Le strutture ottenibili per incollamento di «spazi elementari», come le varietà, i fibrati, le varietà fogliettate, possono essere definite da «atlanti di incollamento» e, formalmente, come categorie arricchite su opportune categorie ordinate.

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Work partially supported by M.P.I. Research Projects.     

Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems

Summary This paper considers the optimal quadratic cost problem (regulator problem) for a class of abstract differential equations with unbounded operators which, under the same unified framework, model in particular «concrete» boundary control problems for partial differential equations defined on a bounded open domain of any dimension, including: second order hyperbolic scalar equations with control in the Dirichlet or in the Neumann boundary conditions; first order hyperbolic systems with boundary control; and Euler-Bernoulli (plate) equations with (for instance) control(s) in the Dirichlet and/or Neumann boundary conditions. The observation operator in the quadratic cost functional is assumed to be non-smoothing (in particular, it may be the identity operator), a case which introduces technical difficulties due to the low regularity of the solutions. The paper studies existence and uniqueness of the resulting algebraic (operator) Riccati equation, as well as the relationship between exact controllability and the property that the Riccati operator be an isomorphism, a distinctive feature of the dynamics in question (emphatically not true for, say, parabolic boundary control problems). This isomorphism allows one to introduce a «dual» Riccati equation, corresponding to a «dual» optimal control problem. Properties between the original and the «dual» problem are also investigated.Research partially supported by the National Science Foundation under Grant NSF-DMS-8301668 and by the Air Force Office of Scientific Research under Grant AFOSR-84-0365.     

Mesures associées aux fonctionnelles additives de Markov. II

Sans résuméEquipe de Recherche n 1 « Processus stochastiques et applications » dépendant de la section n 2 «Théories physiques et probabilités » associée au C.N.R.S.     

Sulla totalità delle curve algebriche tracciate sopra una superficie algebrica

Un riassunto dei principali risultati di questa Memoria trovasi nei «Comptes rendus» del 6 febbraio 1905.     

Sur les opérateurs potentiels au sens de Yosida

Sans résuméEquipe de Recherche n 1 «Processus stochastiques et applications» dépendant de la Section n 1 «Mathématiques, Informatique» associée au C.N.R.S.     

Sui probleimi ellittici non lineari con coefficienti crescenti «rapidamente» o «lentamente»

Summary We study boundary value problems in variational form for nonlinear elliptic operators in the case the nonlinearity is not of polynomial type. We give some examples of applications of abstract theorems for homogeneous problems obtained by other authors (e.g. Donaldson, Grossez) to homogeneous «intermediate» and «mixed» problems. Furthermore we prove some existence theorems for non homogeneous problems.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Existence of weak solutions for some nonlinear problems by the schauder method

Sunto Si studiano alcuni problemi ellittici non lineari con il metodo di Schauder; cio consiste nel « congelare» i coefficienti dell'operatore differenziale e gli eventuali vincoli per dimostrare l'esistenza di una soluzione con un argomento di punto fisso.

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Lavoro eseguito nell'ambito del programma «Metodi variazionali e quasi variazionali per l'equilibrio ed il controllo di sistemi sottoposti a vincoli unilaterali» della Facoltà di Scienze M.F.N. dell'Università di Roma e dello GNAFA-CNR.     

Über das Dämpfungsproblem der mathematischen Physik,mit einer Anwendung auf die Akustik großer Räume

Ohne ZusammenfassungMathematische Bearbeitung eines Vortrages, der in der «Nederlandsche Natuurkundige Vereeniging» zu Amsterdam am 26. Januar 1929 gehalten wurde.     

Bifurcation from 2-dimensional to 3-dimensional invariant tori

Summary The existence of 2 -dimensional invariant tori and their bifurcation in 3-dimensional invariant tori are investigated for a family of (non- hamiltonian) differential sistems in R ⁴.Techniques inspired to the « K.A.M. theory » are used to identify « paths of bifurcation » in the parameters space.Work performed under the auspices of the Italian Council of Research (C.N.R.) and of Ministero della Pubblica Istruzione (M.P.I.).     

Jets de fonctions différentiables sur le dual d'un groupe de Lie nilpotent

Sans résumé«Unité Associée au C.N.R.S. n° 169»     

Partie finie d'un système dynamique et deux nouvelles démonstrations du théorème de Hopf

Sans résuméEquipe de Recherche n1 «Processus Stochastiques et applications» dépendant de la section n1 «Mathématiques, Informatique» associée au C.N.R.S.