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On the boundary behavior of unclosed mappings with the inverse Poletsky inequality

Victoria Desyatka, Evgeny Sevost'yanov

Abstract

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality with an integrable majorant, are open, and discrete. Assume that the image of the boundary of the original domain is finitely connected relative to the mapped domain, and the preimage of the boundary of the latter is nowhere a dense set. Then, under certain conditions on the geometry of these domains, it is proved that the specified mappings have a continuous boundary extension. The result is valid even in a more general form, when the majorant in the inverse Poletsky inequality is integrable over almost all concentric spheres centered at each point. In particular, the obtained results are valid for homeomorphisms as well as for open discrete closed mappings with the appropriate modulus condition.

On the boundary behavior of unclosed mappings with the inverse Poletsky inequality

Abstract

The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion, which has been actively studied recently. We consider mappings of domains of the Euclidean space that satisfy the inverse Poletsky inequality with an integrable majorant, are open, and discrete. Assume that the image of the boundary of the original domain is finitely connected relative to the mapped domain, and the preimage of the boundary of the latter is nowhere a dense set. Then, under certain conditions on the geometry of these domains, it is proved that the specified mappings have a continuous boundary extension. The result is valid even in a more general form, when the majorant in the inverse Poletsky inequality is integrable over almost all concentric spheres centered at each point. In particular, the obtained results are valid for homeomorphisms as well as for open discrete closed mappings with the appropriate modulus condition.
Paper Structure (5 sections, 8 theorems, 88 equations, 7 figures)

This paper contains 5 sections, 8 theorems, 88 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Some information on the theory of liftings and cluster sets
  3. Proof of Theorem \ref{['th3']}
  4. Some examples
  5. On equicontinuity in the closure of a domain

Key Result

theorem 1.1

Let $D$ and $D^{\,\prime}$ be domains in ${\Bbb R}^n,$$n\geqslant 2,$ and let $D$ be a domain with a weakly flat boundary. Suppose that $f$ is open discrete mapping of $D$ onto $D^{\,\prime}$ satisfying the relation (eq2*A) at each point $y_0\in D^{\,\prime}.$ In addition, assume that the following

Figures (7)

  • Figure 1: To the proof of Theorem \ref{['th3']}
  • Figure 2: An open quasiregular mapping that satisfies conditions of Theorem \ref{['th3']}
  • Figure 3: To the proof of Lemma \ref{['lem3']}
  • Figure 4: Formulation of Lemma \ref{['lem4']}
  • Figure 5: The formulation of Lemma \ref{['lem2']}
  • ...and 2 more figures

Theorems & Definitions (14)

  • remark 1.1
  • theorem 1.1
  • corollary 1.1
  • proposition 2.1
  • lemma 2.1
  • proof
  • lemma 5.2
  • proof
  • lemma 5.3
  • proof
  • ...and 4 more