bi-Sobolev同胚弱极限的几乎处处可微性
Differentiability almost everywhere of weak limits of bi-Sobolev homeomorphisms
摘要：
本文研究了双Sobolev同胚的弱极限的可微性。给定$p>n-1$，考虑具有正雅可比算子$J_{f_k} >0$几乎处处和$\sup_k(\|f_{k}\|_{W^{1,n-1}} + \|f_{k}^{-1}\|_{W^{1,p}}) <\infty$的同胚序列$f_k$。我们证明了如果$f$和$h$分别是$f_k$和$f_k^{-1}$的弱极限，且具有正雅可比行列式$J_f>0$和$J_h>0$，则$h(f(x))=x$和$f(h(y))=y$都保持a.e.\并且$f$和$h$几乎处处可微。
英文摘要：
This paper investigates the differentiability of weak limits of bi-Sobolev homeomorphisms. Given $p>n-1$, consider a sequence of homeomorphisms $f_k$ with positive Jacobians $J_{f_k} >0$ almost everywhere and $\sup_k(\|f_{k}\|_{W^{1,n-1}} + \|f_{k}^{-1}\|_{W^{1,p}}) <\infty$. We prove that if $f$ and $h$ are weak limits of $f_k$ and $f_k^{-1}$, respectively, with positive Jacobians $J_f>0$ and $J_h>0$ a.e., then $h(f(x))=x$ and $f(h(y))=y$ both hold a.e.\ and $f$ and $h$ are differentiable almost everywhere.
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