关于Orlicz-Sobolev类中的映射
On mappings in the Orlicz-Sobolev classes
摘要：
首先证明了Orlicz-Sobolev类$W^{1,ϕ}_{\rm loc}$中的开映射在$W^{1,ϕ}_{\rm loc}$上的Calderon型条件下具有全微分a.e.这是Gehring-Lehto-Menchoff在平面上的著名定理和Väisälä在${\Bbb R}^n$，$n\geqslant3$中的著名定理的推广。在同样的条件下，证明了$W^{1,ϕ}_{\rm loc}$中的连续映射$f$，特别是$f\in W^{1,p}_{\rm loc}$对$p>n-1$具有Lusin在a.e.超平面我们的例子证明了Calderon型条件不仅是充分的，而且是必要的，特别地，在a.e.上的$W^{1,n-1}_{\rm loc}$中存在关于$(n-1)$ -维Hausdorff测度不具有$(N)$ -性质的同胚.超平面在此基础上，证明了在此条件下，$ϕ$中的有限变形同胚$f$，特别是$p>n-1$的$f\in W^{1,p}_{\rm loc}$，是$Q(x)$等于其外扩张$K_f(x)$的所谓下$Q$ -同胚，以及与$Q_*(x)=[K_{f}(x)]^{n-1}$的所谓环$Q_*$ -同胚.这使得有可能应用我们的理论的本地和边界行为的较低和环$Q$同胚同胚与有限失真的Orlicz-Sobolev类。
英文摘要：
First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,ϕ}_{\rm loc}$ under the Calderon type condition on $ϕ$ have the total differential a.e. that is a generalization of the well-known theorems of Gehring-Lehto-Menchoff in the plane and of Väisälä in ${\Bbb R}^n$, $n\geqslant3$. Under the same condition on $ϕ$, we show that continuous mappings $f$ in $W^{1,ϕ}_{\rm loc}$, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ have the $(N)$-property by Lusin on a.e. hyperplane. Our examples demonstrate that the Calderon type condition is not only sufficient but also necessary for this and, in particular, there exist homeomorphisms in $W^{1,n-1}_{\rm loc}$ which have not the $(N)$-property with respect to the $(n-1)$-dimensional Hausdorff measure on a.e. hyperplane. It is proved on this base that under this condition on $ϕ$ the homeomorphisms $f$ with finite distortion in $W^{1,ϕ}_{\rm loc}$ and, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms where $Q(x)$ is equal to its outer dilatation $K_f(x)$ as well as the so-called ring $Q_*$-homeomorphisms with $Q_*(x)=[K_{f}(x)]^{n-1}$. This makes possible to apply our theory of the local and boundary behavior of the lower and ring $Q$-homeomorphisms to homeomorphisms with finite distortion in the Orlicz-Sobolev classes.
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