Table of contents

A detailed study of the concept of p-connectedness is carried out; in particular, a criterion for the p-connectedness of two disjoint domains with Lipschitz boundaries and with fractal contact is formulated. New examples of open periodic sets with positive effective conductivity are constructed on the basis of this analysis. A new class of objects, elliptic operators in a Euclidean space with measure, is introduced; the corresponding concept of p-connectedness is introduced and a generalized theory of homogenization is developed.

New natural geometric characteristics are introduced for planar linear trees: the boundary set and the twist number. It turns out that the number of convexity levels of the boundary set is bounded above by a linear function of the twist number. As consequences of this general fact, some non-trivial assertions are obtained about the geometry of linear trees that are extremals of the length or weight functional.

We prove a general theorem that establishes a relation between linear and algebraic independence of values at algebraic points of E-functions and properties of the ideal formed by all algebraic equations relating these functions over the field of rational functions. Using this theorem we prove sufficient conditions for linear independence of values of E-functions as well as for algebraic independence of values of subjects of them. The main result is an assertion stating that at all algebraic points, except finitely many, the values of E-functions are linearly independent over the field of all algebraic numbers if the corresponding functions are linearly independent over the field of rational functions. The theorem is applied to concrete E-functions.

The problem of the limiting distribution of the zeros of the polynomial extremal in the -metric with respect to a measure with finitely many points of growth is studied under the assumption that the degree of this polynomial and the number () of points of growth of the measure approach infinity so that .

A sequence of boundary-value problems for a second-order non-linear elliptic equation in domains and is considered. No geometric assumptions on the  are made. The existence of a sequence approaching zero as is assumed such that for and for an arbitrary point . Here is the -cube with centre at and is the -capacity. The conditions imposed on the coefficients of the equation ensure that the energy space is . The strong convergence of the solutions of the problems under consideration is proved in for ; a corrector in and a homogenized boundary-value problem are constructed. These results are based on an asymptotic expansion for the sequence and on a new pointwise estimate of the solution of a certain model non-linear problem.