On mappings in the Orlicz-Sobolev classes

Abstract

First of all, we prove that open mappings in Orlicz-Sobolev classes $W_{\mathrm{loc}}^{1,\varphi }$ under the Calderon type condition on $\varphi$ 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 ${\mathbb{R}}^n$, $n⩾3$. Under the same condition on $\varphi$, we show that continuous mappings $f$ in $W_{\mathrm{loc}}^{1,\varphi }$, in particular, $f\in W_{\mathrm{loc}}^{1,p}$ 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 $W_{\mathrm{loc}}^{1,n-1}$ 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 $\varphi$ the homeomorphisms $f$ with finite distortion in $W_{\mathrm{loc}}^{1,\varphi }$ and, in particular, $f\in W_{\mathrm{loc}}^{1,p}$ 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 $Q_{\ast }$-homeomorphisms with $Q_{\ast }(x)=[K_f(x){]}^{n-1}$. 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.

69 pages, 2 figures, In Memory of Alberto Calderon (1920-1998)

References in corpus (2)

• On homeomorphisms with finite distortion in the plane
• On equicontinuity of homeomorphisms with finite distortion in the plane

Cited by in corpus (8)

• On finitely Lipschitz space mappings
• On convergence and compactness of space homeomorphisms
• On boundary behavior of spatial mappings
• On $p$-modulus estimates in Orlicz-Sobolev classes
• Convergence and compactness of the Sobolev mappings
• Generalized quasiisometry on smooth riemannian manifolds
• On finite lipschitz Orlicz Sobolev classes
• The theory of prime ends and spatial mappings