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On singularities of mappings with a finite length distortion

Victoria Desyatka, Evgeny Sevost'yanov

TL;DR

The paper investigates removability of isolated boundary singularities for mappings with finite length distortion (FLD) under inverse Poletsky-type modulus inequalities. A central result shows that the continuous extendability to a boundary point $x_0$ follows if a distortion characteristic $Q(y)=K_{I,p}(y,f^{-1},D)$ is integrable on spheres near a cluster point $y_0\in C(x_0,f)$ (and certain multiplicity conditions hold near $y_0$). These results extend quasiregular removability phenomena to FLD mappings and include corollaries for the case $Q\in L^1(\mathbb{R}^n)$. An example demonstrates the necessity of considering the cluster set, showing that integrability on all spheres or extensions beyond the cluster set cannot be generally guaranteed.

Abstract

We study the possibility of a continuous extension of a class of mappings to an isolated point on the boundary of a domain. We show that if some characteristic of this mapping is integrable on almost all spheres in the neighborhood of at least one point of the corresponding cluster set, then this mapping has a continuous extension to the specified point. In particular, this assertion is true if the specified characteristic is simply Lebesgue integrable in the neighborhood of at least one limit point.

On singularities of mappings with a finite length distortion

TL;DR

The paper investigates removability of isolated boundary singularities for mappings with finite length distortion (FLD) under inverse Poletsky-type modulus inequalities. A central result shows that the continuous extendability to a boundary point follows if a distortion characteristic is integrable on spheres near a cluster point (and certain multiplicity conditions hold near ). These results extend quasiregular removability phenomena to FLD mappings and include corollaries for the case . An example demonstrates the necessity of considering the cluster set, showing that integrability on all spheres or extensions beyond the cluster set cannot be generally guaranteed.

Abstract

We study the possibility of a continuous extension of a class of mappings to an isolated point on the boundary of a domain. We show that if some characteristic of this mapping is integrable on almost all spheres in the neighborhood of at least one point of the corresponding cluster set, then this mapping has a continuous extension to the specified point. In particular, this assertion is true if the specified characteristic is simply Lebesgue integrable in the neighborhood of at least one limit point.

Paper Structure

This paper contains 6 sections, 8 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On mappings with inverse Poletsky inequality
  4. On mappings with inverse Poletsky inequality
  5. On removable singularities of $FLD$ mappings
  6. An example

Key Result

theorem 1.1

Let $D$ be a domain in ${\Bbb R}^n,$$n\geqslant 2,$ let $p\geqslant n$ and let $x_0\in D.$ Let $f:D\setminus\{x_0\}\rightarrow {\Bbb R}^n$ be an open discrete mapping with finite length $(p, q)$-dis-tor-ti-on. Let $Q(y):=K_{I, p}\left(y, f^{\,-1}, D\setminus\{x_0\}\right)\geqslant c$ for some $c>0$

Theorems & Definitions (16)

  • remark 1.1
  • theorem 1.1
  • remark 1.2
  • remark 1.3
  • remark 1.4
  • proposition 2.1
  • proposition 2.2
  • lemma 2.1
  • remark 2.5
  • theorem 3.2
  • ...and 6 more