Morse theory and multiple positive solutions to a Neumann problem

Summary We study the equation –du+u=u ^p–1 in a bounded domain , with Neumann boundary conditions. Precisely, we obtain Morse-like relations for the energy functional associated to the problem; as a consequence, we get some multiplicity results when the topology of the boundary is rich.Sponsored by M.U.R.S.T. (fondi 60% «Problemi differenziali nonlineari e teoria dei punti critici»; fondi 40% «Equazioni differenziali e calcolo delle variazioni»).     

Nontrivial solutions of elliptic boundary value problems with resonance at zero

Summary We consider a general nonlinear parabolic BVP (P) on a bounded and smooth domain R_n, the nonlinearity being given by a functionf: . We impose various hypotheses on f: « nonresonance » (with respect to the linearized BVP) at infinity, « nonresonance » or «resonance» at zero. Using an extension of Conley's index theory to noncompact spaces, we prove the existence of equilibria of (P) (i.e. solutions of a corresponding elliptic equation), as well as trajectories joining some of these equilibria. The results obtained generalize earlier results of Amann and Zehnder (who were the first to apply the Conley index to elliptic equations), of Peitgen and Schmitt, and of this author.Dedicated to Professor Jack K. Hale on his 55-th birthdayThis research was supported, in part, by a grant from the Deutsche Forschungsgemeinschaft (D.F.G.).     

Some examples of random walks on free products of discrete groups

Summary We consider the random walk (X_n) associated with a probability p on a free product of discrete groups. Knowledge of the resolvent (or Green's function) of p yields theorems about the asymptotic behaviour of the n-step transition probabilities p^*n(x)=P(X_n= x¦ X₀=e) as n. Woess [15], Cartwright and Soardi [3] and others have shown that under quite general conditions there is behaviour of the type p^*n(x)C_x^– n n^– 3/2. Here we show on the other hand that if G is a free product of m copies ofZ ^r and if (X_n) is the « average » of the classical nearest neighbour random walk on each of the factorsZ ^r, then while it satisfies an « n^–3/2 — law » for r small relative to m, it switches to an n^– r/2 -law for large r. Using the same techniques, we give examples of irreducible probabilities (of infinite support) on the free groupZ ^*m which satisfyn ^– for .     

Temps d'arrÊt d'un système dynamique

Summary The most general positive integer-valued random variable v such that for a given bijective measure preserving transformation , the transformation ^v is still bijective and measure preserving is shown to be a (generally infinite) superposition of return times.

---

---

Equipe de Recherche n 1 «< Processus stochastiques et applications»> dépendant de la section n 2 «Théories Physiques et Probabilités» associée au C.N.R.S.     

OnBMO regularity for linear elliptic systems

Summary We prove a refinement of Campanato's result on local and global (under Dirichlet boundary conditions) BMO regularity for the gradient of solutions of linear elliptic systems of second order in divergence form: we just need that the coefficients are «small multipliers of BMO()», a class neither containing, nor contained in . We also prove local and global L^p regularity: this result neither implies, nor follows by the classical one by Agmon, Douglis and Nirenberg.Work partially supported by M.P.I.Project 40% «Equazioni di evoluzione e applicazioni fisico-matematiche».     

Propriétés relatives des processus de Markov récurrents

Summary For a Markov process recurrent in the sense of Harris, we study the recurrence between two sets A and B. We show the existence of a finite measure _B ^A invariant for the operator P _A P _B and prove some ergodic theorem. When there exists a dual process, _B ^A is the equilibrium measure of B corresponding to the dual process killed when entering in A.

---

---

Equipe de Recherche n 1 «Processus stochastiques et applications», dépendant de la section n 2 «Théories Physiques et Probabilités», associée au CNRS.     

Finite additivity and upper and lower distribution functions

Summary Suppose U is a set and F is a field of subsets of U. Suppose is a real-nonnegative-valued finitely additive function defined on F and is a function from F into a collection of subsets of .An extension of the «usual» notion of distribution function is introduced by defining the notions of upper and lower distribution functions of with respect to . Two fundamental inequalities, for the case of with vounded range union, are shown. These are shown to imply characterization theorems that imply previous integrability and summability theorems (Ann. di Mat. pur e applic., vol. 96, 1973, pp. 265–287)of the author. A measurability (Boll. U.M.I., (4),vol. 7, 1973, pp. 42–49)characterization theorem is shown; this theorem is a consequence of the above mentioned characterization for -integrability, and in turn implies the -summability characterisation. Set function analogues of the «standard» distribution function integral representation theorems are shown.     

Lightlike periodic trajectories in space-times

Let ., _z be a Lorentz metric on a manifold such that isnot compact. We prove the existence of infinitely many lightlike periodic trajectories in by using variational methods and Ljusternik-Schnirelman theory.Sponsored by M.U.R.S.T. (fondi 60% «Problemi differenziali nonlineari e teoria dei punti critici»; fondi 40% «Equazioni differenziali e calcolo delle variazioni»).     

Bornological properties of spaces of non-Archimedean valued functions

Summary Let C(X, K) be the space of continuous functions from an ultraregular space X into a non-Archimedean field K. C(X, K) is a locally convex space with semi-norms ·_F where F belongs to a familyT of bounding subsets of theZ-repletion v₀X of X. We give necessary and sufficient conditions for C(X, K) to be bornological, semi-bornological or ultrabornological.

---

Résumé Soit C(X, K) l'espace des fonctions continues sur l'espace ultraregulière X avec valeurs dans le corps non-Archimédien K. C(X, K) est un espace localement convexe avec semi-normes ·_F où F appartient à une familleT de sous-ensembles bornantes de laZ-replété_v0X de X. Nous donnons des conditions nécessaires et suffisantes pour que X soit bornologigue, semi-bornologique ou ultrabornologique.

---

---

«Aangesteld navorser» of the Belgian «Nationaal Fonds voor Wetenschappelijk Onderzoek».     

Régularisation S.C.I. et γ-convergence: Approximations inf-convolutives associées à un référentiel

Summary The well-known Moreau-Yosida approximation is inoperative for nonconvex functionals without an a priori inequality of quadratic type. We point out an inf-convolution approximation (generally locally lipschitz) by a convex function (called « referential ») connected to the «infinite negative growth» of the approximated function. This method is easily extended to problems of - convergence: epi- convergence, epi-hypo-convergence ... Several applications are given to the case of integral functionals and to conditional expectations of integrands.

---

---

article dédié à J. J.Moreau     

Asymptotic behaviour of nonlinear dispersive models with variable coefficients

We consider mathematical models of evolution which are conservative and include in the simplest case, an equation describing the unidirectional propagation of weakly nonlinear, dispersive long waves suffering disturbances due to the possible unevennes of the botton surface. Our main result gives rates of decay of the amplitude in terms of the «alterations» of the bottom surface.

---

Resumo Consideramos modelos matemáticos de evolução do tipo conservativo sendo que um deles descreve a propagação unidirectional de ondas (fracamente) não- lineares e dispersivas as quais sofrem disturbios devido a possibilidade de o fundo do canal não ser raso. Nosso resultado central neste trabalho estabelece taxas de decaimento da amplitude em termos das «alteraçes» no fundo do canal.

     

Sul criterio dell'energia per la stabilitá di un sistema termomeccanico

Summary The energy criterion for mechanical stability asserts that the stable configurations are those that minimize the potential energy. Recent studies have shown that the energy criterion can be extended to stability of thermomechanical systems under suitable environment conditions, provided that the «stored energy» is interpreted as the equilibrium free-energy at the environmental temperature ^e. The aim of this paper is to provide a contribution to a general theory of thermomechanical stability. Essentially we have restated the theory for general materials introduced by Gurtin with a new framework in the light of recent theories of Noll and Coleman-Owen on simple materials and on thermodynamical potentials. We define a «thermomechanical system» which posseses two main features: i) state space has a «natural topology» depending on the thermodynamical behaviour of system; ii) internal energy E and entropy S are not supposed to exist but are expressely obtained with their smoothness properties.

---

---

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

Portes logiques rapides pour la fonction «ET»

Résumé Spécifications générales du système CLR/A: couplage direct; régime rapide 2ns FWHM/–0.7 V crête sur 75 ; en régime lent (4ns) 0,5 V suffit (entrées), mais 1,2 V/75 est disponible (sorties); étages d'entrée et de sortie standardisés (abrégés en EE et ES). On présente différentes possibilités de connexions entre 3EE et un ES pour réaliser une porte «ET«.ConfigurationA, extrêmement simple, est utilisable en régime lent (4ns),B etC permettent en plus les régimes rapides et mixtes.B est utilisé pour la porte à deux entrées CLR/A 527.C est plus compliqé, mais permet l'extension à quatre canaux (réalisation: CLR/A 502). En appliquant un signal inversé à une entrée, on obtient la fonction «VETO» (anticoïncidence). Sorties normales et inversées sont disponibles, dont chacune peut commander au moins deux autres appareils CLR/A. Temps de passage 6 à 8 ns. Oscillogrammes: (1) réponses ET et à deux trains de 48 et 50 M pulses/sec; (2) réponse ET à deux trains de paires (séparées par 5 ns) aux mêmes taux de répétition qu'en (1); (3) réponses «singles», «doubles» et «triples» en régime mixte sur trois entrées (rapide, semirapide, très lent).     

Propagation of Gevrey singularities for solutions of linear differential operators with constant coefficients

Summary In [4]L. Hörmander has given sufficient conditions for propagation of C singularities for solutions of linear differential operators P with constant coefficients in terms of limit operators called «localization of P at infinity». In this paper a result (Theorem 1.2) of the same type concerning the propagation of Gevrey singularities is given.     

On certain distinguished spectral sets

Sunto Ogni insieme parzialmente ordinato (X, ) ammette varie T₀-topologie «compatibili» con , nel senso che coincide con l'ordine indotto da su X (cf. Hochster [30]). Tra tali topologie la meno fine è la COP(= closures of points) -topologia, cioè la topologia meno fine per la quale l'insieme {y X: xy} è chiuso per ogni x X. La più fine è L(= left)-topologia discreta di Alexandroff, X^L, avente come base per gli aperti gli insiemi {y X: yx} al variare di x in X. In questo lavoro sono date condizioni su X affinchè questo abbia una struttura topologica spettrale noetheriana. Inoltre, vengono caratterizzati gli insiemi parzialmente ordinati X per i quali X^L è uno spazio spettrale; vengono anche caratterizzati gli spazi spettrali Y la cui topologia coincide con la L-topologia associata all'ordine indotto. La topologia dell' «ordine opposto » (opposite-order topology di Hochster, op. cit. Prop. 8) determina una «dualità» tra le L-topologie spettrali e le topologie spettrali noetheriane. Se Y=Spec (A) con la topologia di Zariski, allora Y^H coincide con la O(=ordine) topologia di G. Picavet.

---

---

This work was supported in part by grants from the University of Tennessee at Knoxville and the Istituto Matematico, Università di Roma.

---

This work was performed under the auspices of the GNSAGA of the CNR.

---

This research was funded by a grant from the Research Council of the Graduate School, University of Missouri-Columbia and, also, by the Istituto Matematico, Università di Roma.     

Some dimension-free features of vector-valued martingales

Summary Given any local maringaleM inR ^d orl ², there exists a local martingaleN inR ², such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR ² which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|² and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144     

On the existence of global first integrals in the plane

Summary A plane, autonomous, noncritical differential system x=f(x) of classC ^k is given. Under a suitable «discreteness» hypothesis on the behavior of orbits, the exeistence of a global first integral as regular as the field (provided k )is proved.     

Une démonstration simplifiée et une extension de la formule d'Abramov sur l'entropie des transformations induites

Summary Abramov's formula on the entropy of induced automorphisms is simplified and extended to random powers of automorphisms.

---

---

Equipe de recherche n 1 «< Processus stochastiques et applications»>, dépendant de la section n 2 «, associée au C.N.R.S.     

L r regularity for the stokes and navier-stokes problems

The L^r regularity, 12. In this paper, Cattabriga's results are generalized for W^2, domains. First, we prove a L² regularity result by using appropriate difference quotients of the weak solution; for these, we obtain uniform estimates as a consequence of standard results concerning mixed problems. In order to obtain L^r regularity, we use the hydrodynamical potentials. We deduce L^r «a posteriori» estimates for the strong solution by arguing as D. Gilbarg and N. S. Trudinger have previously done to analyze the Dirichlet problem for the Laplace and Poisson equations. From these results, it is straightforward to demonstrate existence and uniqueness of the strong solution. Also, by applying the usual «boot-strap» argument, one deduces the L^r regularity of any weak solution to the Navier-Stokes problem.

---

Resumé La régularité L^r, 12. Dans ce travail, les résultats de Cattabriga sont généralisés pour des domaines de classe W², . D'abord, afin de démontrer un résultat de régularité L², on considère des quotients différentiels convenables de la solution faible, on analyse des problèmes mixtes dont ces quotients sont les solutions et on en déduit des estimations uniformes. Pour obtenir des résultats de régularité L^r, on utilise les potentiels hydrodynamiques. On obtient des estimations L^r «a posteriori» d'après des arguments analogues aux utilisés par D. Gilbarg et N. S. Trudinger pour le problème de Dirichlet pour les équations de Laplace et Poisson. On en déduit l'existence et l'unicité de solution forte. Aussi, en utilisant la technique «boot-strap», on obtient un résultat de régularité L^r pour toute solution faible du problème de Navier-Stokes.

---

---

This research was partially supported by DGICYT, Spain, Proyecto no. PB92-0696.     

Optimal control problems via a direct method

Summary In this paper we deal with the minimization problem of a cost functional associated to a nonlinear boundary value control problem of a general form, defined in the fixed time interval [0, 1]. Specifically, we first give conditions which ensure that the nonlinear boundary value control problem is solvable and we study the structure of the relative solution set. Then, based on the properties of this set, we establish conditions ensuring both the existence of quasisolutions and that of solutions of the minimization problem under consideration. Such conditions will depend also on the choice of the control space L^r([0, 1],R ^m) where 1 r + Partially supported by the research project M.P.I. (40%) «Teoria del controllo dei sistemi dinamici».