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.

中文翻译：

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约翰域上的 Sobolev 同胚扩展

以平面单位圆盘为源，Jordan 域为目标，我们研究将给定边界同胚扩展为 Sobolev 同胚的问题。对于一般目标，经典 Jordan-Schoenflies 定理的这个 Sobolev 变体可能不承认任何解——它可能有一个边界同胚，它承认一个连续的 $W^{1,2}$-扩展，但甚至不是一个同胚的 $W^ {1,1}$-扩展名。我们证明，如果假设目标是一个约翰圆盘，那么来自单位圆的任何边界同胚允许所有指数 $p<2$ 的 Sobolev 同胚扩展。约翰圆盘是单面的拟定盘，在几何函数理论中具有重要的意义。