Singular asymptotic expansions and delta waves for Burgers' equation

This paper is concerned with the characterization of the weak limits (delta waves) associated to the Cauchy problem for the Burgers' equation and the inviscid Burgers' equation with strongly singular initial data in the form of a regularization by smooth mollifiers of sums of derivatives of Dirac measures. By means of Laplace's method we give the precise asymptotic expansion of the solutionsu in powers of . Then we apply these asymptotics in order to classify completely all possible delta waves under a suitable nondegeneracy condition on some mollifiers regularizing the leading singular term of the initial data. We propose also certain stability results for the weak limits under suitable perturbations of the initial data.Partially supported by 60% of MURST, ItalyPartially supported by grant MM-410/94 with MES, Bulgaria and by 40% of MURST, Italy     

Optimal trajectories of infinite-horizon deterministic control systems

We study the infinite-horizon deterministic control problem of minimizing ₀ ^T L(z, ) dt, T, whereL(z, ·) is convex in for fixedz but not necessarily jointly convex in (z, ). We prove the existence of a solution to the infinite-horizon Bellman equation and use it to define a differential inclusion, which reduces in certain cases to an ordinary differential equation. We discuss cases where solutions of this differential inclusion (equation) provide optimal solutions (in the overtaking optimality sense) to the optimization problem.A quantity of special interest is the minimal long-run average-cost growth rate. We compute it explicitly and show that it is equal to min _x L(x, 0) in the following two cases: one is the scalar casen = 1 and the other is' when the integrand is in a separated form      

On the Existence of a Generalized Solution of One Partial Differential System

We propose a method for the construction of generalized solutions for some nondivergent partial differential systems using set-valued analogs of the generalized statement of the problem based on subdifferential calculus. We establish new sufficient conditions for the existence of solutions of a variational inequality with set-valued operator under weakened coerciveness conditions. We consider examples of a weighted p-Laplacian in the Sobolev spaces , p 2.     

Dolbeault cohomology of compact nilmanifolds

LetM=G/ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.Research partially supported by MURST and CNR of Italy.Research partially supported by MURST and CNR of Italy.     

Properties A and B of nth Order Linear Differential Equations with Deviating Argument

Sufficient conditions for the nth order linear differential equation

On the Asymptotic Behavior of Solutions of Differential Systems

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ (t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality

On Measure-Valued Processes Generated by Differential Equations

We study the problem of representation of a homogeneous semigroup { _t } _{t 0} of transformations of probability measures on in the form where satisfies a differential equation of a special form dependent on the measure . We give necessary and sufficient conditions for this representation.     

On the Solution of the Dirichlet Problem for an Elliptic Equation Degenerating on the Boundary

The Dirichlet problem for the equation is studied in the semi-circle . The restrictions on F are established under which the problem is uniquely solvable in the definite generalized sense.     

Multivalued perturbations for a class of nonlinear evolution equations

We study in this paper the initial value problem for the multivalued differential equation wheref is in MS(2) andG is a multifunction fromC([0, T];) into the closed subsets of L²(0, Y;), satisfying suitable regularity assumptions. As an application we prove a local existence result for the problem

Singular Dirichlet boundary value problems II: Resonance case

Existence results are established for the resonant problem y + _m a y = f(t, y) a.e. on [0, 1] with y satisfying Dirichlet boundary conditions. The problem is singular since f is a Carathéodory function, with a > 0 a.e. on [0, 1] and      

Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem in a smooth boundary domain , as well as the corresponding evolution equation . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L ¹ -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L ¹ -contraction principle for the associated evolution equation.     

Bestimmung und Anwendung von Vekua-Resolventen

The paper is concerned with the differential equation with . The Vekua resolvents are determined by means of an associated second-order differential equation. Applications are given to pseudo-analytic functions, to a differential equation in the theory of several complex variables and to the Ernst equation in general relativity.     

Affine Isometric Embeddings and Rigidity

The Pick cubic form is a fundamental invariant in the (equi)affine differential geometry of hypersurfaces. We study its role in the affine isometric embedding problem, using exterior differential systems (EDS). We give pointwise conditions on the Pick form under which an isometric embedding of a Riemannian manifold M ³ into is rigid. The role of the Pick form in the characteristic variety of the EDS leads us to write down examples of nonrigid isometric embeddings for a class of warped product M ³'s.     

Smooth Solution of the Dirichlet Problem for a Quasilinear Hyperbolic Equation of the Second Order

On the basis of the exact solution of the linear Dirichlet problem , we obtain conditions for the solvability of the corresponding Dirichlet problem for the quasilinear equation u _tt – u _xx = f(x, t, u, u _t).     

Cohomology of the Vector Fields Lie Algebra and Modules of Differential Operators on a Smooth Manifold

Let M be a smooth manifold, the space of polynomial on fibers functions on T*M (i.e., of symmetric contravariant tensor fields). We compute the first cohomology space of the Lie algebra, Vect(M), of vector fields on M with coefficients in the space of linear differential operators on . This cohomology space is closely related to the Vect(M)-modules, (M), of linear differential operators on the space of tensor densities on M of degree .     

On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let be a family of k-subsets of . The -shadow of is the set of all (k-)-subsets contained in the members of . Let be a t-intersecting family (any two members have at least t elements in common) with . Given k,t,m the problem is to minimize (over all choices of ). In this paper we solve this problem when m is big enough.     

A sharp existence and localization theorem for a Neumann problem

We shall present a new version of a recently appeared theorem for the existence and localization of solutions of the Neumann problem associated to the equation , based on a general variational principle by Ricceri. Our study will be especially aimed to express a certain hypothesis regarding the function g in its sharpest form, and a limit case is enquired by an approximation Received: 7 July 2003     

BRST Operator for Quantum Lie Algebras and Differential Calculus on Quantum Groups

For a Hopf algebra , we define the structures of differential complexes on two dual exterior Hopf algebras: (1) an exterior extension of and (2) an exterior extension of the dual algebra ^*. The Heisenberg double of these two exterior Hopf algebras defines the differential algebra for the Cartan differential calculus on . The first differential complex is an analogue of the de Rham complex. When ^* is a universal enveloping algebra of a Lie (super)algebra, the second complex coincides with the standard complex. The differential is realized as an (anti)commutator with a BRST operator Q. We give a recursive relation that uniquely defines the operator Q. We construct the BRST and anti-BRST operators explicitly and formulate the Hodge decomposition theorem for the case of the quantum Lie algebra U _q(gl(N)).     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

Increasing Monotone Operators in Banach Space

An operator mapping a separable reflexive Banach space X into the dual space X is called increasing if as . Necessary and sufficient conditions for the superposition operators to be increasing are obtained. The relationship between the increasing and coercive properties of monotone partial differential operators is studied. Additional conditions are imposed that imply the existence of a solution for the equation with an increasing operator A.