Table of contents

An explicit expression is obtained for the operator solving the Stokes problem in terms of operators solving the Dirichlet problems for the Poisson and Laplace equations. With the help of this expression a theorem is proved on a sharp estimate of the remainder in the asymptotics of the eigenvalues of the Stokes operator. Bibliography: 13 titles.

Let and be symmetric operators in a Hilbert space , such that is positive and has an arbitrary spectrum. In this paper nonhomogeneous boundary value problems are considered for an equation of the form

(1)

An abstract theorem (of the Lax-Milgram type) is proved, which is then used to prove theorems on the weak and strong solvability of boundary value problems for equation (1) in the energy spaces defined by the operators and , as well as a theorem on the traces of a strong solution. As an application, nonhomogeneous boundary value problems for partial differential equations are considered. Bibliography: 16 titles.

Theorems on regularization and solvability in Sobolev spaces are established for equations whose principal symbol may have stationary points of contact type. Bibliography: 14 titles.

One possible method is suggested for the constructive description of function classes defined on continua of the complex plane for which the traditional (in this subject area) description in terms of distances from boundary points to corresponding level curves of the outer Riemann function generally does not exist. The main idea of the constructions and results presented consists in taking account of the growth of derivatives of polynomials approximating the function. Bibliography: 28 titles.

For the equation , where is a homogeneous positive polynomial of degree , and , the first boundary value problem is considered in a conical domain. The asymptotics of the solution at infinity is studied under the condition that the right side and the boundary functions asymptotically coincide with polynomials. Bibliography: 7 titles.

The authors prove a theorem which characterizes the limit distribution of the zeros of polynomials , , defined by one (for each ) extremal relation with a variable (depending on ) weight function. Bibliography: 9 titles.

Let be a free group and . The structure of the group is studied. A description is obtained of the periodic part of in terms of the third and fourth homology groups of . It is shown that, if is residually torsion-free nilpotent, the same is true for the factor group of by its periodic part. Bibliography: 9 titles.

The method of Fourier series for entire and meromorphic functions was developed by Rubel and Taylor. Rubel conjectured that similar results are valid for subharmonic functions in , , and suggested the use of spherical harmonics. In this paper a positive solution is given to this conjecture. As corollaries, many-dimensional analogues of classical theorems on entire functions due to Weierstrass, Borel and Lindelöf are deduced. Bibliography: 23 titles.

The author defines the algebraic tensor structures which characterize the maximally mobile spaces of hyperplane elements with a general affine connection, and establishes the maximal orders of the groups of motions G_r in these spaces. The main aim is to determine a tensor test for a maximally mobile space. Bibliography: 8 titles.

Let be an arbitrary bounded proper continuum on , a finite collection of pairwise distinct domains that are components of , a function meromorphic in each domain and continuous in some neighborhood of , the sum of the principal parts of the Laurent expansions of with respect to its poles in the union of the domains in , and the degree of the rational function . If all the domains are bounded, then . If is a rectifiable curve , then the total variation of along satisfies

where is the supremum of the set of total variations along of all the partial fractions with . Bibliography: 11 titles.

This article gives an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property. Bibliography: 37 titles.

An expression in the form of a Riemann theta-function is obtained for the asymptotic behavior of polynomials orthogonal on a system of contours. Properties of a limit-periodic discrete Sturm-Liouville operator and the dynamics of a periodic Toda lattice are considered as a consequence of the asymptotic formulas obtained. Figures: 2. Bibliography: 21 titles.

The author describes all pairs (l, m), where m is the minimal number of generators of the group of units of a finite commutative local ring, and l is the dimension of its field of residues over its prime subfield. Bibliography: 3 titles.

This article gives a necessary and sufficient condition for a function which is holomorphic in a neighborhood of zero to belong to the class . This criterion, which is formulated in terms of the Taylor coefficients of the function, is then applied to give a description of the singular set of holomorphic functions of several variables which admit rapid rational approximation relative to Lebesgue measure (i.e., which belongs to the class ). In particular, Theorem.If , then the complement of the envelope of holomorphy is a pluripolar set. This theorem together with a well-known result of A. A. Gonchar gives a complete description of the domains for which : this property is satisfied if and only if is a pluripolar set. Bibliography: 11 titles.

The concept of splitting a homotopy equivalence along a system of submanifolds is introduced, and examples are given in which such a splitting is not possible. Bibliography: 6 titles.