Toward the theory of Orlicz–Sobolev classes

Abstract:

It is shown that, under a Calderón type condition on the function $\varphi$, the continuous open mappings that belong to the Orlicz–Sobolev classes $W^{1,\varphi }_{\mathrm {loc}}$ have total differential almost everywhere; this generalizes the well-known theorems of Gehring–Lehto–Menchoff in the case of ${\mathbb R}^2$ and of Väisälä in ${\mathbb R}^n$, $n\geq 3$. Appropriate examples show that the Calderón type condition is not only sufficient but also necessary. Moreover, under the same condition on $\varphi$, it is also proved that the continuous mappings of class $W^{1,\varphi }_{\mathrm {loc}}$ and, in particular, of class $W^{1,p}_{\mathrm {loc}}$ for $p>n-1$ have Lusin’s $(N)$-property on a.e. hyperplane. On that basis, it is shown that, under the same condition on $\varphi$, the homeomorphisms $f$ with finite distortion of class $W^{1,\varphi }_{\mathrm {loc}}$ and, in particular, those belonging to $W^{1,p}_{\mathrm {loc}}$ for $p>n-1$, are what is called lower $Q$-homeomorphisms, where $Q$ is equal to their outer dilatation $K_f$; also, they are so-called ring $Q_*$-homeomorphisms with $Q_*=K_{f}^{n-1}$. The latter fact makes it possible to fully apply the theory of the boundary and local behavior of the ring and lower $Q$-homeomorphisms, as developed earlier by the authors, to the study of mappings in the Orlicz–Sobolev classes.