A classification of almost contact metric manifolds

Summary It is obtained a complete classification for almost contact metric manifolds through the study of the covariant derivative of the fundamental 2- form on those manifolds.This work was supported by the « Consejería de Educación del Gobierno de Canarias ».     

Second order abstract differential equations with singular coefficients

Summary We study a large class of second order linear abstract differential equations, whose coefficients can be singular. In the framework of suitable « weighted » spaces, we prove some existence and uniqueness results for generalized and ordinary solutions of initial value problems for such equations.This work was partially supported by the G.N.A.F.A. and the Istituto di Analisi Numerica of the C.N.R. (Italy).     

On some abstract variable domain hyperbolic differential equations

The Cauchy problem is studied for a class of linear abstract differential equations of hyperbolic type with variable domain. Existence and uniqueness results are proved for (suitably defined) weak solutions. Some applications to P.D.E. are also given: they concern linear hyperbolic equations either in non-cylindrical regions or with mixed variable lateral conditions.This work was supported in part by the M.U.R.S.T. (Italy), through 60% and 40% research funds, and by the «G.N.AF.A. of the C.N.R.» (Italy).     

Derivations of differential forms on jet bundles

Summary The theory of derivations of differential forms, originally formulated by Frölicher and Nijenhuis [2], is generalized to include derivations which connect differential forms on two different manifolds. This generalized theory is then used to analyze derivations of direct limits of exterior algebras of forms on jet bundles. Such derivations have found applications in the analysis of Lie equations [3], [4] and in the inverse problem of the calculus of variations [6], [7], [8], [9].

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Résumé On présente une généralization de la théorie des dérivations des formes extérieures, formulée par Frölicher et Nijenhuis [2], au cas des dérivations qui relient les formes extérieurs sur deux variétés différentes. On utilize ensuite cette théorie généralizée pour étudier les dérivations des limites directes des algèbres des formes extérieures sur les fibrés des jets. Ces dérivations ont été appliquées dans l'analyse des équations de Lie [3], [4] et dans le problème inverse du calcul des variations [6], [7], [8], [9].

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This work is a part of a program conducted jointly with ProfessorBenenti at Istituto di Fisica Matematica «J. L. Lagrange» of Torino. The authors are greatly indebted to the Director of the Institute Professor D.Galletto for his interest and encouragement.This work has been supported by Gruppo Nazionale per la Fisica Matematica del Consiglio Nazionale delle Ricerche.     

The universal connection of an arbitrary system

Using the theory of smooth spaces, we generalize the notion of finite dimensional system of connections on a fiber bundle to the concept of arbitrary system of connections. Then we study the universal connection of a regular system and the universal curvature.This work has been performed during the visit of Prof. I.Kolá at Department of Applied Mathematics «G. Sansone», University of Florence, supported by GNSAGA of CNR. The second author has been supported by the grant N. 201/93/2125 of GA CR. This work has also been partially supported by local and national funds of MURST.     

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.     

The local structure of trans-Sasakian manifolds

Summary In this paper, we completely characterize the local structure of trans-Sasakian manifolds of dimension 5 by giving suitable examples.Work supported by the «Consejería de Educación del Gobierno de Canarias».     

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.).     

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.

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article dédié à J. J.Moreau     

Some results on surface measures in calculus of variations

We study some relations between the concepts of perimeter, Hausdorff measure, and Minkowsky content, when R ^N is endowed with a convex Finsler metric depending in a continuous way on the position. We show some connections with the theory of -convergence and with the anisotropic motion of a smooth hypersurface by mean curvature.This work was partially supported by NSF Grant DMS-9008999, and by MURST (Progetto Nazionale «Equazioni di Evoluzione e Applicazioni Fisico-Matematiche» and «Analisi Numerica e Matematica Computazionale») and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.     

Submanifolds of generalized Hopf manifolds,type numbers and the first Chern class of the normal bundle

Summary Any orientable real hypersurface M of a complex Hopf manifold (carrying the locally conformal Kaehler (l.c.K.) metric discovered by I.Vaisman [33]) has a natural f-structure P as a generic Cauchy-Riemann submanifold; we show (cf. our § 5) that if P anti-commutes with the Weingarten operator, then the type number of the hypersurface is less equal than 1. Moreover, M carries the natural almost contact metrical structure observed by Y.Tashiro [30]; if its almost contact vector is an eigenvector of the Weingarten operator corresponding to a nowhere vanishing eigenfunction and the holomorphic distribution is involutive, then M is foliated with globally conformai Kaehler manifolds (cf. our § 5), provided that some restrictions on the type number of M are imposed. We derive (cf. our § 6) a «Simons type» formula and apply it to compact orientable hypersurfaces with non-negative sectional curvature (in a complex Hopf manifold) and parallel mean curvature vector. Several examples of submanifolds of l.c.K. manifolds are exhibited in § 3. Our § 7 studies complex submanifolds of generalized Hopf manifolds; for instance, we show that the first Chern class of the normal bundle of a complex submanifold having a flat normal connection is vanishing.     

Green's function and invariant density for an integro-differential operator of second order

Summary We construct the Green function for a parabolic- elliptic integro- differential operator of second order with boundary conditions given by a first order operator, both with Hölder continuous coefficients. Also, we prove the existence of a unique invariant density associated with the corresponding reflected diffusion process with jumps. This allows us to study the asymptotic behaviour of the solution for some oblique derivative problems.This research have been supported in part by Ministero P.I. « Progetto Caleolo delle Variazioni » and N.S.F. under grant No. DMS-8702236.     

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     

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.     

On some degenerate evolution variational inequalities

Summary A class of parabolic variational inequalities is investigated, where: the unilateral constraints concern the time derivative of the unknown function; degeneracies or singularities with respect to the time variable are considered. Some general abstract existence and uniqueness results are proved, in the natural framework of suitable weighted spaces.

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Sunto Si studia una classe di disequazioni variazionali paraboliche, dove: i vincoli unilaterali riguardano la derivata rispetto al tempo delta funzione incognita; vengono considerate singolarità e degenerazioni rispetto alla variabile temporale. Si dimostrano alcuni risultati generali di esistenza ed unicità, nell'ambito naturale di opportuni spazi con peso.

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This work was supported in part by the «G.N.A.F.A. del C.N.R.», and by the «Ministero dell'Università e della Ricerca Scientifica» (through 60% and 40% grants).     

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 ».     

Symmetrization in a nonlinear degenerate parabolic problem

Summary Bounds for the solution of a nonlinear degenerate parabolic problem are given by means of the solution of a «symmetrized» problem.This work was performed under a national program of Italian M.P.I. (60%, 1985).     

Experimentelle Untersuchungen über den Einfluss der Eintrittsgrenzschicht auf die Druckumsetzung in geraden und gekrümmten Diffusoren

Ohne ZusammenfassungEin ausführlicher Bericht erscheint in den «Mitteilungen aus dem Institut für Aerodynamik an der ETH», herausgegeben von Prof. Dr.J. Ackeret.     

Rami di varietà lungo sottovarietà da un punto di vista topologico

Summary A topological definition is given of « branch of a complex algebraic variety along a subvariety ». Comparisons are made with classical algebraic branch theory.     

Existence and uniqueness for a reaction-diffusion problem in infiltration

We consider a system of coupled PDE'smodeling the infiltration of a reacting fluid in a soluble porous medium. The system is made of a parabolic equation for the concentration of the dissolved material, an ODE (hyperbolic equation with characteristic x=Const.)for the porosity, and an elliptic equation for the fluid pressure. We prove the existence and uniqueness of a classical solution. The classical solution is global in time in the one-dimensional case. Global existence of a weak solution is proved for the n- dimensional case.The authors would like to acknowledge the M.U.R.S.T. Project 40% «Problemi non lineari...» and the Italia C.N.R. Strategic Project «Metodi matematici per le applicazioni industriali» for partial financial support of this work.