Sobolev homeomorphic extensions onto John domains
Sobolev homeomorphic extensions onto John domains
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DOI：
10.1016/j.jfa.2020.108719
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发表时间：
2020-04
期刊：
影响因子：
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通讯作者：
P. Koskela;Aleksis Koski;Jani Onninen
中科院分区：
文献类型：
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作者：
P. Koskela;Aleksis Koski;Jani Onninen
以平面单位圆盘为源，Jordan域为目标，研究了将给定的边界同胚扩张为Sobolev同胚的问题。对于一般的目标，这个经典的Jordan-Schöenflies定理的Sobolev变体可能不承认任何解-它可能有一个边界同胚，它允许一个连续的W 1，2-扩张，但甚至不允许一个同胚的W 1，1-扩张。我们证明了如果目标是一个John圆盘，那么单位圆上的任何边界同胚都有一个Sobolev同胚扩张，且指数p< 2。John圆是单侧拟圆，在几何函数论中具有重要意义.
Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schöenflies theorem may admit no solution-it is possible to have a boundary homeomorphism which admits a continuous W 1, 2-extension but not even a homeomorphic W 1, 1-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents p< 2. John disks, being one sided quasidisks, are of fundamental importance in Geometric Function Theory.