Table of contents

Two groups of singularly perturbed boundary value problems for eigenvalues are considered. The first group contains the following types of perturbations: a small parameter in front of the leading derivatives in the equation or in the boundary condition, a domain with rapidly oscillating boundary, or a thin domain. Problems with perturbations of the data on a set with small diameter ε constitute the second group.

Theorems on the existence of a solution are proved for problems of control of unstable semilinear evolution and elliptic equations. Problems of distributed control and boundary control with distributed and terminal observation are studied.

This article is an investigation of a circle of questions connected with optimal recovery of a function in a class of functions with smoothness determined by a differential operator. Linear methods of recovery are considered.

A characterization is given for an m-meromorphic continuation of a continuous function f on the segment [-1,1], in terms of the asymptotic behavior of the finite poles of the mth row of the table of Tchebycheff rational approximants to f.

Regular dependence theorems are presented and growth estimates in Hölder norms are established for the solutions of boundary value problems, under perturbation of the domain by vector fields. The question of parametrization of domains is considered. The results obtained are applied to the analysis of operator functions which arise in the study of a free boundary problem.

This paper gives an explicit classification of the maximal arithmetic subgroups of the group of K-points of an indefinite orthogonal K-group of type (D_l), where K is a global field, charK ≠ 2.

It is proved that if is a nonatomic measure space and an even function nondecreasing on and such that , for , and for all , , then the space does not contain boundedly compact Tchebycheff sets with more than one point.

New estimates are given in the two-dimensional case for special operators that are linear combinations of Riesz transforms. They are used to investigate the distances between partial derivatives , , on the class

The situation is considered when either the transitional or initial measure is vector-valued (the other is, respectively, scalar-valued; thus the product measure is also vector-valued). The integrable function is vector-valued. In this situation two theorems of Fubini type are proved.

It is proved that a solution of the Boltzmann equation in the limit case of scattering by small angles tends to a solution of the Landau-Fokker-Planck equation.

The author studies questions on the existence and uniqueness of generalized solutions of a problem with incomplete initial conditions for symmetric degenerate hyperbolic systems of first order with possible growth of the coefficients of the spatial part. Sharp theorems are proved for the existence and uniqueness of generalized solutions in classes of functions which become unbounded on approach to regions of degeneracy and singularity.

Conditions are obtained for a classical solution on a contact manifold when the initial motion of the free boundary is equal to zero.

The author exhibits an application of ideal theory to the exact integration of trigonometric polynomials with arbitrarily prescribed coefficient spectrum and to the approximate integration of functions of certain multidimensional classes E, W, and H.

Some geometric properties of projective spaces over the classical fields R, C, and H are studied, which yield a new description of their symplectic cobordism modules. These results lead to new relations in the symplectic cobordism ring of a point. As an application, it is proved that θ₁θ_iθ_j = 0, where the θ_i are Ray's elements.

The Navier-Stokes equations

are considered on a two-dimensional closed manifold imbedded in . Theorems on existence and uniqueness of generalized solutions of steady-state and time-dependent problems are proved. Unique solvability of the Euler equations () is proved by passing to the limit as . The existence of a maximal attractor for the Navier-Stokes system on is proved, and for the case where the manifold is the sphere an estimate for the Hausdorff dimension of the attractor is obtained:

The multiplicative structure is computed for the bordism ring MSU_* of manifolds with stable SU-structure in the stable normal bundle.

Conditions for the representation of arbitrary analytic functions in the form of an exponential series are studied.

This article is devoted to a study of the boundary behavior of solutions of the Dirichlet problem for the biharmonic equation in arbitrary plane regions.