Table of contents

The Stefan problem for a quasilinear parabolic equation is considered. Convective motions in the fluid phase are described by the Navier-Stokes system. The existence of a smooth solution locally in time is proved. Bibliography: 15 titles.

The problem of approximating functions by trigonometric polynomials with a given number of harmonics is considered. The exact degree of approximation is determined for three closely related classes of smooth functions: those defined by differential properties, by difference properties, and functions in Besov spaces. In contrast to the classical case, the degrees of approximation for these classes turn out to be different from each other. Bibliography: 14 titles.

Inequalities of the form

(1)

are studied, where , and are positive monotone functions, and denotes, respectively, a) a multidimensional Calderón-Zygmund singular integral extended over a domain in ( is the distance from to the boundary of the domain); and b) the conjugate function (, ). In case a) a class of domains is distinguished (domains of type in ) which, in particular, contains domains with smooth boundaries; for each domain of type , , sufficient conditions are found for the validity of (1), and examples are given which demonstrate their necessity. In case b) we give necessary and sufficient conditions for the validity of (1). For monotone weight functions these results amplify and supplement corresponding investigations by Hunt, Muckenhoupt, and Wheeden (Trans. Amer. Math. Soc.176 (1973), 227-251) and by Coifman and Fefferman (Studia Math.51 (1974), 241-250). Bibliography: 32 titles.

Conditions are established under which the solution of the first boundary value problem for a sequence of linear or quasilinear uniformly elliptic equations with weakly convergent coefficients converges to the solution of the respective limit problem. One of the main requirements in those conditions is weak equicontinuity, with respect to the independent variables, of the leading coefficients of the equations being considered. Examples show that these conditions of the theorems are essential. Bibliography: 18 titles.

Groups of the form are studied, where is the free product of groups , , and is the th term of the derived series of the Cartesian subgroup of this product. It is proved that if every is conjugacy separable, residually finite with respect to occurrence in cyclic subgroups, and torsion-free, then the groups are conjugacy separable. Bibliography: 8 titles.

It is proved that all differential operators of the form

(1)

whose spectrum coincides with the spectrum of the linear oscillator

(2)

i.e. , , and whose potentials are sufficiently smooth and differ sufficiently little from the potential may be obtained by the well-known method of the theory of the inverse Sturm-Liouville problem. This result was obtained earlier by McKean and Trubowitz (MR 83e:34034). This paper gives another proof of this theorem, based on the following completeness theorem, which is interesting in itself. Denote by the eigenfunctions of equation (1) and by the eigenfunctions of equation (2). The linear span of the set of functions

is dense in the space . Bibliography: 8 titles.

This article is devoted to questions of the uniqueness of multiple trigonometric series. Uniqueness theorems are obtained for multiple trigonometric series with additional hypotheses on their coefficients or spherical Abel means, and also on their Lebesgue means. Bibliography: 19 titles.

Normal forms of fibrations of binomial surfaces are found. The results are applied to the study of slow motions of equations of relaxation type. Figures: 2. Bibliography: 7 titles.

For the Weil-Petersson metric on the Teichmüller space of marked Riemann surfaces of genus 0 with punctures, a potential is constructed in terms of the density of the hyperbolic metric on the corresponding surface (i.e., in terms of a solution of Liouville's equation). It is shown that this potential is a generating function of the accessory parameters of the Fuchsian uniformization of the corresponding Riemann surface. Also, a connection is established between the accessory parameters and the Eichler integrals of Fuchsian groups. Bibliography: 18 titles.

The transforms

are introduced for an integer and a given vector . Their duality is substantiated, applications of the differentiation operations are studied, and other properties of -transforms are established. A number of examples are given to illustrate the method of -transforms for solving some classes of differential equations and boundary value problems for partial differential equations. Bibliography: 9 titles.

Scattering of a system of several particles in an external electric field is considered. In particular, completeness of wave operators is proved for a system of several particles in a strong electric field. The main attention is devoted to a system of three particles, and in this case the completeness of wave operators is proved. Bibliography: 26 titles.

The author determines conditions under which a connected unramified holomorphic covering of a Liouville space is again a Liouville space. Bibliography: 31 titles.

A theorem is proved on stochastic stability of Lyapunov exponents of almost all linear differential equations of arbitrary order. Bibliography: 16 titles.

The Euler expansion of Dirichlet series constructed from the Fourier coefficients of Siegel modular forms for principal congruence subgroups of the integral symplectic group is obtained. Bibliography: 6 titles.

This article is a study of sufficiently general necessary conditions for optimality in a time-optimal problem with fixed endpoints and with control appearing in a linear manner. Bibliography: 6 titles.

The author considers the validity of an estimate in the norm of the Hölder spaces for the solutions of linear elliptic equations , where for all (, ). This estimate does not depend on the smoothness of the coefficients . It is known (MR 83c:35059) that such an estimate holds for sufficiently small exponents depending on and . In this paper it is proved that this dependence is essential: for every one can exhibit a constant and construct a sequence in of elliptic equations, of the indicated form with smooth coefficients, whose solutions converge uniformly in the unit ball to a function that does not belong to . Bibliography: 5 titles.