Український математичний вісник, 2016, № 1

Метрические свойства классов Орлича-Соболева

2016-01-01T00:00:00Z

Метрические свойства классов Орлича-Соболева Салимов, Р.Р. В работе изучаются гомеоморфизмы классов Орлича–Соболева при условии типа Кальдерона на функцию φ. Для таких отображений установлен целый ряд теорем о локальном поведении и, в частности, доказан аналог известной теоремы Геринга о локальной липшицевости, приведены различные теоремы об оценке искажения евклидовых расстояний.; In this article, we consider the homeomorphisms of the Orlicz-Sobolev class under a condition of the Calderon type on φ. For these classes of mappings, a number of theorems on the local behavior are established, and, in particular, an analog of the famous Gehring theorem on a local Lipschitz property as well as various theorems on estimates of distortion of the Euclidean distance are proved.

Solutions of some partial differential equations with variable coefficients by properties of monogenic functions

2016-01-01T00:00:00Z

Solutions of some partial differential equations with variable coefficients by properties of monogenic functions Pogorui, A.A.; Rodriguez-Dagnino, R.M. In this paper we study some partial differential equations by using properties of Gateaux differentiable functions on commutative algebra. It is proved that components of differentiable functions satisfy some partial differential equations with coefficients related with properties of bases of subspaces of the corresponding algebra.

Adapted statistical experiments

2016-01-01T00:00:00Z

Adapted statistical experiments Koroliouk, D.V. We study statistical experiments with random change of time, which transforms a discrete stochastic basis in a continuous one. The adapted stochastic experiments are studied in continuous stochastic basis in the series scheme. The transition to limit by the series parameter generates an approximation of adapted statistical experiments by a diffusion process with evolution.

On a model semilinear elliptic equation in the plane

2016-01-01T00:00:00Z

On a model semilinear elliptic equation in the plane Gutlyanskii, V.Y.; Nesmelova, O.V.; Ryazanov, V.I. Assume that Ω is a regular domain in the complex plane C and A(z) is symmetric 2 × 2 matrix with measurable entries, det A = 1 and such that 1/K|ξ|² ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|², ξ ∊ R², 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = e^u in Ω and show that the well-known Liouville–Bieberbach function solves the problem under an appropriate choice of the matrix A(z). The proof is based on the fact that every regular solution u can be expressed as u(z) = T(ω(z)) where ω : Ω → G stands for quasiconformal homeomorphism generated by the matrix A(z) and T is a solution of the semilinear weihted Bieberbach equation ∆T = m(w)e^T in G. Here the weight m(w) is the Jacobian determinant of the inverse mapping ω⁻¹(w).