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.     

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.).     

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 ».     

Asymptotically linear elliptic problems at resonance

Summary In this paper we study the existence of solutions for some semilinear, elliptic equation with homogeneous Dirichelet boundary conditions and a non linear term which is asymptotically linear «at resonance».In memoria di Paolo Bartolo     

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.).     

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.     

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.     

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).     

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.     

On the direct solution of Riccati equations arising in boundary control theory

Two classes of Riccati equations arising in the boundary control of parabolic systems are studied by direct methods. The new feature with respect to previous works on this subject is the low regularity of the final data. The classes considered here generalize those of [7]and [5]on one side, and of [14]on the other one. Completely new methods are used to obtain the solution of the Riccati equations, in both cases. The central theme is the dependence of the solutions on a «symmetric» norm of the final data, yielding these new results as well as a new proof of existence for the related algebraic Riccati equation under more general assumptions. The synthesis of the associated linear-quadratic-regulator problems is easily solved using these results.     

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.     

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»).     

Weierstrass multiple points and ramification points of smooth projective curves

Summary In this paper we study finite sets of smooth algebraic curves which are the support of special divisors («Weierstrass sets»). We prove several existence results of Weierstrass sets with low weight on suitable curves (e.g. general k-gonal curves).     

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».     

On families of singular plane projective curves

Summary In this article it is studied, in modern form, the classical concept of «complete continuous system of projective plane curves of degree n, with singularities of a certain type». Properties of these schemes are studied (universal properties, techniques to compute tangent spaces, smoothness, special points, etc.). As an application, a modern presentation of the classical theories of systems of curves with nodes, and with nodes and cusps, is given.     

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.     

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,     

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.     

Grassmann bundles

Summary In the classical theory of the Grassmann Variety there are three principal results. The Basis Theorem asserts that the Chow ring has a selfdual linear basis of classes. Determinantal Formulawhich expresses any basic class as a determinant in the special classes. Finally the ring structure is elucidated by Pieri's Formulawhich expresses the intersection of a basic class and a special class in terms of the basic classes. Here we show how all these results can be established also for the Chow ring of a Grassmann bundle. There are however some differences. In the classical case the basic classes are Schubert classes: this is impossible in the general case as there need not be enough Schubert classes to provide a basis and in the general case there is a pair of dual bases which both reduce to the Schubert basis in the classical case. In addition to these generalizations of the classical results we also enlarge on the theory of Schubert classes developed in the important paper of Kempf and Laksov [4].Following them we shall henceforth use the phrase « determinantal formula » to mean their formula for Schubert classes and our generalization of it to « improper » Schubert classes.     

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).