Sobolev Jordan-Schonflies问题
The Sobolev Jordan-Schonflies Problem
摘要：
以平面单位圆盘$\mathbb D$为参考构形，Jordan域$\mathbb Y$为变形构形，研究了将给定的边界同胚$φ\colon \partial \mathbb D \to \partial \mathbb Y$扩张为复平面的Sobolev同胚的问题.研究经典Jordan-Schönflies定理的Sobolev变体是由非线性弹性理论（NE）和几何函数理论（GFT）中相关纯位移变分问题的适定性所激发的。显然，边界映射$φ$允许$W^{1,p}$ -Sobolev同胚扩张的必要条件是它首先允许连续的$W^{1,p}$ -Sobolev扩张。然而，对于任意靶结构域$\mathbb Y$，这是不够的。
英文摘要：
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.
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