论文标题
Sobolev同质形态扩展到John域
Sobolev homeomorphic extensions onto John domains
论文作者
论文摘要
鉴于平面单位磁盘是源和约旦域作为目标，我们研究了将给定边界同态形态扩展为Sobolev同态的问题。对于一般目标，这种经典的Jordan-Schoenflies定理的Sobolev变体可能承认没有解决方案 - 有可能具有边界同构性，该边界同构承认连续的$ W^{1,2} $延伸，甚至没有同源$ W^{1,1,1} $ - 扩展。我们证明，如果假定目标是John Disk，那么单位圈的任何边界同态性都承认所有指数的Sobolev同型扩展名$ P <2 $。 John Disks是单方面的准风险，在几何函数理论中至关重要。
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-Schoenflies 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.
