Table of contents

We consider a Dirichlet problem in which the boundary value of a solution is understood as the -limit, , of traces of this solution on surfaces 'parallel' to the boundary. We suggest a setting of this problem which (in contrast to the notion of solution in ) enables us to study the solvability of the problem without making smoothness assumptions on the coefficients inside the domain. In particular, for an equation in selfadjoint form without lower-order terms, under the same conditions as those used for , we prove unique solvability and establish a bound for an analogue of the area integral.

Bibliography: 37 titles.

The paper is devoted to a topological analysis of the Kovalevskaya-Yehia integrable case in rigid body dynamics. It is proved that the integral has the Bott property on isoenergy surfaces of the system; the topology of the Liouville foliation in a neighbourhood of degenerate 1-dimensional orbits and equilibria (points of rank 0) is also described. In particular, marked loop molecules are constructed for degenerate 1-dimensional orbits, and a representation in the form of an almost direct product is found for nondegenerate singularities of rank 0.

Bibliography: 17 titles.

Given a linear constant-coefficient elliptic equation of arbitrary order on a two-dimensional strip, a criterion is obtained for the existence of the mean-square limits of its solutions on the boundary of the strip.

Bibliography: 2 titles.

The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found.

Bibliography: 9 titles.

The following result is proved: if approximations in the norm of  (of ) of functions in the classes (in , respectively) by some linear operators have the same order of magnitude as the best approximations, then the set of norms of these operators is unbounded. Also Bernstein's and the Jackson-Nikol'skiǐ inequalities are proved for trigonometric polynomials with spectra in the sets (in ).

Bibliography: 15 titles.

Simple methods are used to give new proofs, and sometimes to make them more precise, of basic theorems on isometric surfaces with a common mean curvature, which are usually called Bonnet pairs. The considerations are conducted under the assumption of minimally admissible smoothness of the objects in question, and certain necessary or sufficient criteria are given for the non-existence of Bonnet pairs with a common non-constant mean curvature among compact surfaces.

Bibliography: 26 titles.