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On boundary non-preserving mappings with integral constraints

Victoria Desyatka, Oleksandr Dovhopiatyi, Evgeny Sevost'yanov

Abstract

This manuscript is devoted to the study of mappings, satisfying the upper weighted Poletsky inequality. We study the case where the boundary of the domain may not be preserved under the mapping and, besides that, the majorant from the above inequality satisfies constraints of the integral-type. Under certain additional conditions on the definition domain and the corresponding cluster sets, we prove that families of above mappings are equicontinuous in the closure of this domain.

On boundary non-preserving mappings with integral constraints

Abstract

This manuscript is devoted to the study of mappings, satisfying the upper weighted Poletsky inequality. We study the case where the boundary of the domain may not be preserved under the mapping and, besides that, the majorant from the above inequality satisfies constraints of the integral-type. Under certain additional conditions on the definition domain and the corresponding cluster sets, we prove that families of above mappings are equicontinuous in the closure of this domain.
Paper Structure (4 sections, 11 theorems, 92 equations)

This paper contains 4 sections, 11 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['th1']}
  4. Sobolev and Orlicz-Sobolev classes

Key Result

theorem 1.1

Let $p\geqslant 1,$ let $D$ be a domain in ${\Bbb R}^n,$$n\geqslant 2.$ Assume that: 1) the set $E$ is nowhere dense in $D,$ and $D$ is finitely connected on $E,$ i.e., for any $z_0\in E$ and any neighborhood $\widetilde{U}$ of $z_0$ there is a neighborhood $\widetilde{V}\subset \widetilde{U}$ of $z holds for some $\delta_0>\tau_0:=\Phi(0).$ Let the family of all components of $D^{\,\prime}_f\setm

Theorems & Definitions (13)

  • theorem 1.1
  • lemma 2.1
  • proposition 2.1
  • lemma 2.2
  • remark 2.1
  • lemma 4.3
  • proposition 4.2
  • lemma 4.4
  • lemma 4.5
  • lemma 4.6
  • ...and 3 more