Capacity estimates, Liouville’s theorem, and singularity removal for mappings with bounded (p, q)-distortion

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• 作者：A. N. Baykin ; S. K. Vodop’yanov
• 关键词：mappings with bounded weighted (p ; q) ; distortion ; capacity estimate ; Liouville ; type theorem ; singularity removal
• 刊名：Siberian Mathematical Journal
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：56
• 期：2
• 页码：237-261
• 全文大小：364 KB
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• 作者单位：A. N. Baykin (1) (2)
  S. K. Vodop’yanov (1) (2)
  
  1. Sobolev Institute of Mathematics, Moscow, Russia
  2. Novosibirsk State University, Novosibirsk, Russia
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Russian Library of Science
  
• 出版者：Springer New York
• ISSN：1573-9260

文摘

The mappings with bounded weighted (p, q)-distortion are natural generalizations of the class of mappings with bounded distortion which appears as a doubly indexed scale for p = q = n in the absence of weight functions. In case n ?1 < q ?p = n, the mappings with bounded (p, q)-distortion were studied previously in a series of articles under the additional assumption that the mapping enjoys Luzin’s N-property. In this article we present the first facts of the theory of mappings with bounded (p, q)-distortion which are obtained without additional analytical assumptions. The core of the theory consists of the new analytical properties of pushforward functions; in particular, we prove that the gradient of the pushforward function vanishes almost everywhere on the image of the branch set. Some estimates are given on the capacity of the images of condensers under mappings with bounded (p, q)-distortion. We obtain Liouville-type theorems and the singularity removal theorems for the mappings of this class, and we apply these theorems to classifying manifolds.