论文标题
Sobolev Jordan-Schonflies问题
The Sobolev Jordan-Schonflies Problem
论文作者
论文摘要
我们将平面单元盘$ \ Mathbb D $视为参考配置，而Jordan域$ \ Mathbb Y $是变形的配置，并研究了扩展给定边界同质形态$φ\ colon \ colon \ colon \ colon \ colon \ partial \ pottial \ pottial \ partbb d \ to partial \ partbb to \ partbb y mathbb y $ as s s s osbolev as s s osbolev的问题。在非线性弹性理论（NE）和几何函数理论（GFT）中，研究了经典Jordan-Schönflies定理的这种Sobolev变体的这种sobolev变体的动机。显然，边界映射$φ$接受A $ W^{1，p} $ - SOBOLEV同型扩展的必要条件是，它首先承认连续$ w^{1，p} $ - sobolev扩展。但是，对于任意目标域$ \ mathbb y $，这是不够的。
We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $φ\colon \partial \mathbb D \to \partial \mathbb Y$ as a Sobolev homeomorphism of the complex plane. Investigating such a Sobolev variant of the classical Jordan-Schönflies theorem is motivated by the well-posedness of the related pure displacement variational questions in the theory of Nonlinear Elasticity (NE) and Geometric Function Theory (GFT). Clearly, the necessary condition for the boundary mapping $φ$ to admit a $W^{1,p}$-Sobolev homeomorphic extension is that it first admits a continuous $W^{1,p}$-Sobolev extension. For an arbitrary target domain $\mathbb Y$ this, however, is not sufficent.
