On a class of Sobolev scalar products in the polynomials

This paper discusses Sobolev orthogonal polynomials for a class of scalar products that contains the sequentially dominated products introduced by Lagomasino and Pijeira. We prove asymptotics for Markov type functions associated to the Sobolev scalar product and an extension of Widom's Theorem on the location of the zeroes of the orthogonal polynomials. In the case of measures supported in the real line, we obtain results related to the determinacy of the Sobolev moment problem and the completeness of the polynomials in a suitably defined weighted Sobolev space.     

A moment problem for one-dimensional nonlinear pseudoparabolic equation

We are concerned with a moment problem for a nonlinear pseudoparabolic equation with one space dimension on an interval. The boundary conditions are imposed in terms of the zero-order moment and the first-order moment. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in the usual Sobolev space. We are able to get regularity of the solution so that both solution and its derivative with respect to the time variable belong to the same Sobolev space with respect to the space variable. This feature is different from problems with parabolic equations, where the regularity order of solution is higher than that of the time derivative with respect to the space variable. Previous results reflected only this parabolic nature for the pseudoparabolic equation.     

The Moment Problem for a Sobolev Inner Product

The concepts of definite and determinate Sobolev moment problem are introduced. The study of these questions is reduced to the definiteness or determinacy, respectively, of a system of classical moment problems by means of a canonical decomposition of the moment matrix associated with a Sobolev inner product in terms of Hankel matrices.     

Existence of solutions to BVP for linear differential-functional equations of elliptic type

We consider a homogeneous boundary value problem for a linear partial differential-functional equation of elliptic type. We prove the existence of a unique solution to such problem in a Sobolev space.     

Notes on limits of Sobolev spaces and the continuity of interpolation scales

We extend lemmas by Bourgain-Brezis-Mironescu (2001), and Maz'ya-Shaposhnikova (2002), on limits of Sobolev spaces, to the setting of interpolation scales. This is achieved by means of establishing the continuity of real and complex interpolation scales at the end points. A connection to extrapolation theory is developed, and a new application to limits of Sobolev scales is obtained. We also give a new approach to the problem of how to recognize constant functions via Sobolev conditions.

     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.     

Stochastic impulse control of parabolic systems of Sobolev type

We consider the optimal impulse control problem for a system whose dynamics is described by a stochastic differential equation of Sobolev type. The coefficients of the equation are closed operators acting in Hilbert spaces. The system is parabolic by virtue of a bound imposed in the right half-plane on the resolvent of the characteristic operator pencil. The results are applied to stochastic partial differential equations of Sobolev type.     

Cheeger Type Sobolev Spaces for Metric Space Targets

In this paper, we consider the natural generalization of Cheeger type Sobolev spaces to maps into a metric space. We solve Dirichlet problem for CAT(0)-space targets, and obtain some results about the relation between Cheeger type Sobolev spaces for maps into a Banach space and those for maps into a subset of that Banach space. We also prove the minimality of upper pointwise Lipschitz constant functions for locally Lipschitz maps into an Alexandrov space of curvature bounded above.     

On the periodic KdV equation in weighted Sobolev spaces

We prove well-posedness results for the initial value problem of the periodic KdV equation as well as Kam type results in classes of high regularity solutions. More precisely, we consider the problem in weighted Sobolev spaces, which comprise classical Sobolev spaces, Gevrey spaces, and analytic spaces. We show that the initial value problem is well posed in all spaces with subexponential decay of Fourier coefficients, and ‘almost well posed’ in spaces with exponential decay of Fourier coefficients.     

The Lazer-McKenna conjecture for an elliptic problem with critical growth, part II

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain.     

On the solvability of a spatial problem of Darboux type for the wave equation

The question of the correct formulation of one spatial problem of Darboux type for the wave equation has been investigated. The correct formulation of that problem in the Sobolev space has been proved for surfaces having a quite definite orientation on which are given the boundary value conditions of the problem of Darboux type.     

On solvability of the second boundary value problem in a half-space for one elliptic equation

We consider the second boundary value problem for one elliptic equation with a lower term in a half-space. We prove theorems of solvability of the problem in the Sobolev space.     

Quasielliptic operators and Sobolev type equations

We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.     

On the solvability of one boundary value problem for some semilinear wave equations with source terms

In a conic domain of time type for one class of semilinear wave equations with source terms we consider a Sobolev problem representing a multidimensional version of the Darboux second problem. The questions on global and local solvability, uniqueness and absence of solutions of this problem are investigated.     

Quasielliptic operators and Sobolev type equations. II

We consider matrix quasielliptic operators on the whole space. Under the quasihomogeneity condition for symbols, we establish the isomorphism theorem for these operators in the special scales of Sobolev spaces. In particular, this result implies a series of available isomorphism theorems for elliptic operators and theorems about the unique solvability of the initial value problem for a broad class of systems of Sobolev type.     

On a coupled nonlinear singular thermoelastic system

For a coupled nonlinear singular system of thermoelasticity with one space dimension, we consider its initial boundary value problem on an interval. For one of the unknowns a classical condition is replaced by a nonlocal constraint of integral type. Because of the presence of a memory term in one of the equations and the presence of a weighted boundary integral condition, the solution requires a delicate set of techniques. We first solve a particular case of the given nonlinear problem by using a functional analysis approach. On the basis of the results obtained and an iteration method we establish the well-posedness of solutions in weighted Sobolev spaces.     

On the well-posedness of the prediction-control problem for certain systems of equations

We consider the inverse problem for equations of Sobolev type and their applications to linearized Navier-Stokes systems and phase-field systems. We obtain conditions for the well-defined solvability of these systems.     

Nonstationary Maxwell equations

ABSTRACT

The paper deals with a mixed problem for nonstationary generalised Maxwell equations. The boundary conditions are of Riemann-Hilbert type. The problem is reduced to a mixed problem for a wave equation where the boundary conditions are of Dirichlet type as they were introduced by D. Spencer in the middle 1950?s. We use the Fourier method to construct an approximate solution to the problem in certain function spaces of Sobolev type.     

Solvability of Nonlocal Problems for Systems of Sobolev-Type Differential Equations with a Multipoint Condition

Russian Mathematics - We consider a nonlocal problem for a system of loaded differential equations of the Sobolev type with a multipoint constraint. By introducing additional unknown functions, we...     

Multielement equations for analytic functions in the plane with cuts

We consider linear equations for analytic functions in the complex plane with cuts along a half of the boundary of a quadrangle. We propose a regularization method that reduces the equations to an equation with summary-difference kernels. Some applications are given to the moment problem for entire functions of exponential type.