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     

A mathematical analysis of quantum transport in three-dimensional crystals

Summary We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain inR ³ (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.     

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.     

Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group ℍ n

In this article we study sets in the (2n + 1)-dimensional Heisenberg group ℍ ⁿ which are critical points, under a volume constraint, of the sub-Riemannian perimeter associated to the distribution of horizontal vector fields in ℍ ⁿ .We define a notion of mean curvature for hypersurfaces and we show that the boundary of a stationary set is a constant mean curvature (CMC) hypersurface. Our definition coincides with previous ones. Our main result describes which are the CMC hypersurfaces of revolution in ℍ ⁿ .The fact that such a hypersurface is invariant under a compact group of rotations allows us to reduce the CMC partial differential equation to a system of ordinary differential equations. The analysis of the solutions leads us to establish a counterpart in the Heisenberg group of the Delaunay classification of constant mean curvature hypersurfaces of revolution in the Euclidean space. Hence, we classify the rotationally invariant isoperimetric sets in ℍ ⁿ .     

Two remarks on Monge-Ampere equations

Summary We consider real Monge-Ampère equations and we present two new properties of these equations. First, we show the existence of the «first eigenvalue of Monge-Ampère equation» i.e. we show the existence of a positive constant possessing all the properties of the first eigenvalue of a 2-nd order elliptic operator (positivity, uniqueness of the eigenfunction, maximum principle, bifurcation...).The second property concerns variational characterisations of solutions. Both properties are closely related to similar properties of the general class of Hamilton-Jacobi-Bellman equations.     

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.     

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.     

Chow groups and Borel-Moore schemes

Summary In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme Cop^k P ⁿ parametrizing finite coplanary subschemes of lenght k ofP ⁿ and of the variety of «complete S-tuples» of Le Barz.The authors were partially supported by the DGICYT.     

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.     

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.     

Planforms in three dimensions

The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are calledplanforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE inR ² (see [6, 7]), In this article we attempt to give such a classification for Euclidean equivariant systems of PDE inR ³. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found.We attempt to find the planforms on all lattices inR ³ that are forced to exist near a steady-state bifurcation from a trivial solution. The proof of our classification uses Liapunov-Schmidt reduction with symmetry (which can be used if we assume spatial periodicity of the solutions) and the Equivariant Branching Lemma. The analytical problem of finding planforms for systems of PDE is reduced to the algebraic problem of computing isotropy subgroups with one dimensional fixed point subspaces.The Navier-Stokes equations and reaction-diffusion equations (with constant diffusion coefficients) are examples of systems of PDE that satisfy the conditions of our classifications. In this article, we show that our classification applies to the Kuramoto-Sivashinsky equation.     

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»).     

Periodic solutions of a class of nonlinear evolution equations

Summary In this paper we prove the existence of periodic solutions of abstract evolution equations which are modelled after parabolic problems. More precisely we prove that existence results follow from degree type hypotheses on the «projection» of the problem onto a suitable finite dimensional space.     

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.     

三维系统中一族闭轨在周期扰动下的分支

讨论一类三维系统在周期扰动下的分支问题.假设此三维系统有一族闭轨,利用 Poincar\'e映射及积分流形定理,得到了在周期扰动下由这族闭轨产生次调和解和不变环面的条件,并讨论了次调和解的鞍结点分支.     

On Lower Dimensional Invariant Tori in C^d Reversible Systems

In this paper, a result on the persistence of lower dimensional invariant tori in Cd reversible systems is obtained under some conditions. The theorem is proved for any d which is larger than some constants.     

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).     

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.

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

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»).     

0-dimensional subschemes ofP 2: New application of Castelnuovo's function

Summary This note first reformulates and sharpens certain newly discovered properties of the first difference of the postulation of a 0-dimensional subscheme ofP ², a function which it seems historically appropriate to call “Castelnuovo's Function?. Those results are then applied in two ways: inverting Bezout's Theorem; settling the existence problem for finite pointsets inP ² with postulation and ?Cayley-Bacharach Number? prescribed. During much of this work the author enjoyed the financial support of CNR (Italy) and the hospitality of the Departments of Mathematics of the Universitics of Ferrara and Genova and the Politecnico of Torino.