Orlicz-Sobolev classes and lower \(Q\)-homeomorphisms (Q2880009)

It is shown that continuous mappings \(f\in W_{\text{loc}}^{1, \varphi}\), where \(\varphi\) satisfies a Calderon type condition, have the \((N)\)-property on a.e. hyperplane. It is proved that under these conditions on \(\varphi\), the mappings \(f\) with finite distortion correspond to the so-called lower \(Q\)-homeomorphisms, where \(Q\) equals the outer dilatation \(K_f\). This makes it possible to apply the theory of local and boundary behavior of lower and ring \(Q\)-homeomorphisms which was introduced by the authors in earlier publications to homeomorphisms with finite distortion in Orlicz-Sobolev classes. More precisely, Theorem 1 states that a function \(f:\Omega\rightarrow {\mathbb R}\) is differentiable a.e. in the open set \(\Omega\subset {\mathbb R}^n\), \(n\geq 3\), provided that \(f\in W_{\text{loc}}^{1, \varphi}\) and \(\varphi\) satisfies the Calderon condition. An analogous result for open mappings \(f:\Omega\rightarrow {\mathbb R}^n\) is stated as Theorem 2. For the Hausdorff measure \(H^k\) and a measurable set \(E\subset \Omega\), Theorem 3 gives a distortion condition for \(H^k(f(E))\) when \(f\in W_{\text{loc}}^{1, \varphi}\) and \(\varphi\) satisfies the Calderon condition. Finally, given domains \(D\) and \(D^{\prime}\subset {\mathbb R}^n\), \(n\geq 3\), it is proved that every homeomorphism \(f:D\rightarrow D^{\prime}\) of finite distortion in the class \(f\in W_{\text{loc}}^{1, \varphi}\) is a lower \(Q\)-homeomorphism at every point \(x_0\in \overline{D}\) provided that \(\varphi\) satisfies the Calderon condition and \(Q\) equals the dilatation \(K_f\). The results obtained in the work are applicable to many problems in geometrical function theory.