Ruslan Salimov

The Poletskii and Väisälä inequalities for the mappings with (p, q)-distortion

This article is devoted to the study of space mappings which are more general than quasiregular. Some modulus inequalities for this class of mappings are obtained. In particular, analogs of the well-known Poletskii and Väisälä inequalities were proved.

Boundary behavior and the Dirichlet problem for Beltrami equations

We show that arbitrary homeomorphic solutions to the Beltrami equations with generalized derivatives satisfy certain moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equations in arbitrary Jordan domains and in arbitrary finitely connected domains bounded by mutually disjoint Jordan curves, correspondingly.

ACL and differentiability of the open discrete ring mappings

We study the so-called ring Q-mappings which are the natural generalization of quasiregular mappings. It is proved that open discrete ring Q-mappings are differentiable a.e. and belong to the class ACL in ℝ n , n ≥ 2; furthermore, provided that .

On convergence analysis of space homeomorphisms

Various theorems on convergence of general spatial homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring Q-homeomorphisms are obtained. In particular, it is established that a family of all ring Q-homeomorphisms f in ℝn fixing two points is compact provided that the function Q is of finite mean oscillation. The corresponding applications have been given to mappings in the Sobolev classes Wloc1,p for the case p > n − 1.

Extension of the Schwarz Lemma to homeomorphisms with controlled p-module

Abstract We extend the classical Schwarz Lemma to homeomorphisms with integrally bounded p-module in ℝn.

Logarithmic Hölder continuity of ring homeomorphisms with controlled -module

We study homeomorphisms preserving integrally quasiinvariant the weighted p-module of the ring domains and provide a condition ensuring the local Hölder continuity of such mappings with respect to logarithms of the distances. The inequality defining the continuity is sharp with respect to the order.

Boundary behavior of Orlicz-Sobolev classes

It is proved that homeomorphisms of the Orlicz-Sobolev class Wloc1, φ can be continuously extended to the boundaries of some domains if the function φ defining this class satisfies a Carderón-type condition and the outer dilatation Kf of the mapping f satisfies the divergence condition for integrals of special form. In particular, the result holds for homeomorphisms of the Sobolev classes Wloc1,1 with Kf ∈ Llocq for q > n − 1.

Regularity of Mappings with Integrally Restricted Moduli

We consider certain classes of homeomorphisms of domains in \(\mathbb R^n\) with integrally bounded p-moduli of the families of curves and surfaces, which essentially extend the well-known classes of mappings such as quasiconformal, quasiisometric, Lipschitzian, etc. In the paper we survey the known results in this field regarded to the differential properties of such homeomorphisms, but mainly present a wide range of open related problems.

On local properties of spatial generalized quasi-isometries

An upper bound for the measure of the image of a ball under mappings of a certain class generalizing the class of branched spatial quasi-isometries is determined. As a corollary, an analog of Schwarz’ classical lemma for these mappings is proved under an additional constraint of integral character. The obtained results have applications to the classes of Sobolev and Orlicz–Sobolev spaces.

Finite mean oscillation in upper regular metric spaces

There are established some sufficient conditions for boundary homeomorphic extension in metric spaces in which the measure of a ball of radius ε is controlled from above by a wide class of functions depending on ε. We consider a class of mappings whose ring moduli are integrally majorated. These results involve a finite mean oscillation and asymptotic estimation of majorants.