On mappings in the Orlicz-Sobolev classes

Abstract

First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,\phi}_{\rm loc}$ under the Calderon type condition on $\phi$ have the total differential a.e. that is a generalization of the well-known theorems of Gehring-Lehto-Menchoff in the plane and of V\"ais\"al\"a in ${\Bbb R}^n$, $n\geqslant3$. Under the same condition on $\phi$, we show that continuous mappings $f$ in $W^{1,\phi}_{\rm loc}$, 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 $\phi$ the homeomorphisms $f$ with finite distortion in $W^{1,\phi}_{\rm loc}$ 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.