Abstract

This article is devoted to the study of mappings with bounded and finite distortion, as well as the Sobolev classes that have been actively studied recently. We study the homeomorphisms of the Sobolev classes \(W^{1,p}_{\textrm{loc}}\) for the case when \(p=n-1\), where \(n\) is the corresponding dimension of space. For these classes, we prove the estimate of the distortion of the modulus of families of curves under mapping. As an application, we obtain results on the boundary extension of homeomorphisms of the indicated class in terms of prime ends.

中文翻译：

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Sobolev类别具有临界指数的Sobolev类的边界扩展

摘要

本文致力于有界和有限失真的映射的研究，以及最近已经积极研究的Sobolev类。对于\（p = n-1 \）的情况，我们研究Sobolev类\（W ^ {1，p} _ {\ textrm {loc}} \）的同胚性，其中\（n \）是对应的空间尺寸。对于这些类，我们证明了映射下曲线族的模量的失真估计。作为应用，我们获得了关于指定类同质素的素数边界扩展的结果。