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On boundary Hölder continuity of Sobolev and Orlicz-Sobolev classes

Victoria Desyatka, Zarina Kovba, Evgeny Sevost'yanov

Abstract

We investigate distortion estimates for mappings at boundary points of a domain. We consider mappings of the Sobolev and Orlicz-Sobolev classes and some other classes of mappings that do not preserve the boundary of a domain. For above mappings, we establish distortion estimates at boundary points. In particular, under certain conditions on the characteristics of the mappings, we show that they are Hölder continuous. In the manuscript we consider both the case of ``good'' boundaries and ``domains with prime ends''. We have obtained not only Hölder-type estimates, but also some more general ones under appropriate (more general) conditions on the characteristic.

On boundary Hölder continuity of Sobolev and Orlicz-Sobolev classes

Abstract

We investigate distortion estimates for mappings at boundary points of a domain. We consider mappings of the Sobolev and Orlicz-Sobolev classes and some other classes of mappings that do not preserve the boundary of a domain. For above mappings, we establish distortion estimates at boundary points. In particular, under certain conditions on the characteristics of the mappings, we show that they are Hölder continuous. In the manuscript we consider both the case of ``good'' boundaries and ``domains with prime ends''. We have obtained not only Hölder-type estimates, but also some more general ones under appropriate (more general) conditions on the characteristic.

Paper Structure

This paper contains 6 sections, 19 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Lemmas
  4. Proof of Theorems \ref{['th4']} and \ref{['th5']}
  5. Some another distance distortion estimates
  6. On classes of ring $Q$-mappings

Key Result

theorem 1.1

Let $D$ be a bounded domain in ${\Bbb R}^n,$$n\geqslant 2,$$x_0\in \partial D.$ 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 4) If $n\geqslant 3,$ assume that, the function $\varphi$ satisfies Calderon condition Then there

Theorems & Definitions (25)

  • theorem 1.1
  • theorem 1.2
  • lemma 2.1
  • lemma 2.2
  • lemma 2.3
  • proposition 2.1
  • lemma 2.4
  • proof
  • proposition 2.2
  • proposition 2.3
  • ...and 15 more