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On boundary extension of unclosed Orlicz-Sobolev mappings

Victoria Desyatka, Evgeny Sevost'yanov

TL;DR

This work addresses boundary extension questions for unclosed Orlicz-Sobolev mappings with distortion controlled by $K_{I,\alpha}$ and bound $Q$. By establishing a Main Lemma under Calderón-type conditions on $\varphi$ and integral/mean-oscillation constraints on $Q$, it proves continuous extension to boundary points and, when these hold on the entire boundary, a global extension $\overline f$ with $\overline f(\overline D)=\overline{D'}$. It further develops equicontinuity results for families of such mappings in the closure, employing modulus, capacity, and liftings arguments, and provides planar and equi-uniform domain variants. These results extend boundary regularity theory to unclosed Orlicz-Sobolev mappings with finite distortion and yield practical criteria for boundary extendability and stability of families under the chordal metric.

Abstract

This paper is devoted to the study of the boundary behavior of Orlicz-Sobolev classes that may not preserve the boundary under mapping. Under certain conditions, we show that these mappings have a continuous extension to the boundary of definition domain.

On boundary extension of unclosed Orlicz-Sobolev mappings

TL;DR

This work addresses boundary extension questions for unclosed Orlicz-Sobolev mappings with distortion controlled by and bound . By establishing a Main Lemma under Calderón-type conditions on and integral/mean-oscillation constraints on , it proves continuous extension to boundary points and, when these hold on the entire boundary, a global extension with . It further develops equicontinuity results for families of such mappings in the closure, employing modulus, capacity, and liftings arguments, and provides planar and equi-uniform domain variants. These results extend boundary regularity theory to unclosed Orlicz-Sobolev mappings with finite distortion and yield practical criteria for boundary extendability and stability of families under the chordal metric.

Abstract

This paper is devoted to the study of the boundary behavior of Orlicz-Sobolev classes that may not preserve the boundary under mapping. Under certain conditions, we show that these mappings have a continuous extension to the boundary of definition domain.

Paper Structure

This paper contains 5 sections, 16 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Lemma
  4. Some additional information
  5. The equicontinuity in the closure of a domain

Key Result

theorem 1.1

Let $\alpha\geqslant 1,$ let $D$ and $D^{\,\prime}$ be bounded domains in ${\Bbb R}^n,$$n\geqslant 3,$ let $b\in \partial D,$ let $Q:D\rightarrow [0, \infty]$ be a Lebesgue measurable function and let $\varphi:[0,\infty)\rightarrow[0,\infty)$ be increasing function. Let $f:D\rightarrow D^{\,\prime}$ Suppose that $K_{I, \alpha}(x, f)\leqslant Q(x)$ a.e. and at least one of the following conditions

Theorems & Definitions (20)

  • theorem 1.1
  • theorem 1.2
  • lemma 2.1
  • lemma 2.2
  • lemma 2.3
  • proposition 2.1
  • lemma 2.4
  • proposition 2.2
  • lemma 3.5
  • proof
  • ...and 10 more