Table of contents

In this paper local properties of weak solutions of nonlinear elliptic systems arising in problems of the deformation theory of plasticity are investigated. -estimates are obtained for a weak solution in the case of plasticity with power-type consolidation. For linear consolidation various properties are established, such as the Hölder continuity of a weak solution, the square-integrability of its second order derivatives, and -estimates for these derivatives. Here the elasticity and plasticity domains are introduced. In the former the solution is regular, while in the latter, when there are more than two variables, a weak solution has Hölder continuous first derivatives in a subdomain that differs from the plasticity domain by a set of measure zero. Bibliography: 20 titles.

It is shown that each algebraic action of a simply connected reductive algebraic group on an affine algebraic variety can be contracted (in a flat one-dimensional family of actions) to a canonical action of on a certain affine variety having some very special properties. It is shown that and have many algebro-geometric properties in common. As an application, we prove the Procesi-Kraft conjecture to the effect that the singularities of the closures of orbits in the case of spherical stabilizer are rational. It is assumed that the ground field has characteristic zero. Bibliography: 37 titles.

This paper is devoted to a generalization of a classical inequality: let be bounded and analytic in the disk ; then , in the case of nonanalytic functions . More precisely, it is proved that if , where is the boundary function of a function of bounded characteristic, and is a function in a quasianalytic class (defined by some condition of regularity of decrease of its Fourier coefficients), then . The proof of this result depends in an essential way on a theorem of Levinson and Cartwright. At the same time, the result strengthens the Levinson-Cartwright theorem. Bibliography: 7 titles.

The problem of scattering by a one-dimensional periodic lattice with impurity potential is considered. A stationary scattering matrix is constructed on the basis of the asymptotics of the scattered waves, its properties are studied, and it is shown to coincide with the nonstationary scattering operator defined in the usual way in the quasimomentum representation of the unperturbed operator . The inverse scattering problem is also solved, i.e., the problem of recovering on the basis of and the scattering data. Here we follow the scheme going back to the well-known work of V. A. Marchenko and L. D. Faddeev. However, solution of the inverse problem in the presence of a periodic lattice required considerable modification of classical arguments. The theory of so-called "global" quasimomentum serves as analytic basis. Conditions on the scattering data are found which are necessary with a finite second moment and sufficient in order that there exist a unique impurity potential with given scattering characteristics and a finite first moment. Bibliography: 29 titles.

Some results are obtained on perturbation of a polyharmonic operator by potentials having small support. Bibliography: 9 titles.

In the case of formally Hamiltonian systems a certain class of statistical solutions which it is natural to call equilibrium solutions is singled out. The properties of these solutions are studied. If the system is sufficiently regular, then each equilibrium solution satisfies the Kubo-Martin-Schwinger condition in the classical form. Bibliography: 15 titles.

The Milnor K-groups of complete discrete valuation fields are studied. Bibliography: 9 titles.

The Rallis mapping is used to obtain commutation formulas for the Hecke operators of the symplectic and orthogonal groups on the space of theta-series. These formulas are applied to the multiplicative arithmetic of representations of quadratic forms by forms. Bibliography: 15 titles.

For any prime a classification (up to similarity) is given of all invariant integral lattices that correspond to an orthogonal decomposition of the Lie algebra . Even unimodular lattices without roots are distinguished. For they contain the Leech lattice. For some of the resulting lattices the automorphism groups are studied, and lower bounds for the minimal length of vectors are obtained. Figures: 2. Bibliography: 17 titles.

A special approach to Riesz potentials on Lorentz spaces is considered. The main properties of these potentials are established. An analogue of the Hardy-Littlewood-Sobolev theorem is proved for them. Finally, applications to the Cauchy problem for the Euler-Poisson-Darboux equation on Lorentz spaces are given. Bibliography: 10 titles.

The equations of the Zassenhaus variety of a classical semisimple Lie algebra over an algebraically closed field of characteristic are found under the condition that does not divide the order of the Weyl group of this algebra. The set of singular points of this variety is described. Bibliography: 9 titles.

First, the questions of existence, multiplicity, and stability of timeperiodic solutions of the van der Pol equations with small diffusion are considered. It is shown that in some situations, the principle of averaging for parabolic equations plays a significant role in justifying these results. In this connection, the justification of the principle is given. At the end of the paper, it is indicated that the results allow one to investigate the well-known problem of the existence of spatially nonhomogeneous regimes in homogeneous media. Bibliography: 26 titles.

Let be a bounded convex domain lying in the left-hand half-plane, with . A class , consisting of functions analytic in and satisfying the inequalities

is said to be quasianalytic at if contains no functions that vanish with all their derivatives at . Let and , , and let

It is shown that the condition

where is the trace function of the sequence , and is the inverse of , is necessary and sufficient for the quasianalyticity of . This theorem generalizes the classical Denjoy-Carleman theorem. In the case when the theorem follows from Salinas's results of 1955. For the theorem was proved by Korenblyum [Korenblum] in 1965. Bibliography: 9 titles.

Existence and uniqueness theorems are established for a generalized solution of a mixed problem for the nonlinear Schrödinger equation in the presence of dissipation in the space and . The method of proving uniqueness of a solution is based on the assumption of the existence and boundedness in of the integral of a solution for some , where is the degree of nonlinearity in the equation. Bibliography: 16 titles.

It is proved that the upper semicontinuity of the central exponents of various classes of linear systems of differential equations with bounded coefficients is typical. Bibliography: 6 titles.