arXiv:1012.5010v2 [math.CV] 25 Dec 2010 On homeomorphisms with finite distortion in the Orlicz-Sobolev classes D. Kovtonyuk, V. Ryazanov, R. Salimov and E. Sevost’yanov IN MEMORY OF ALBERTO CALDERON (1920–1998) November 12, 2018 (OS-251210-ARXIV.tex) Abstract It is developed the theory of the local and boundary behavior of the mappings with finite distortion in the Orlicz-Sobolev classes W 1,ϕ loc and, in particular, in the Sobolev classes W 1,p loc given in domains of R n , n  3, extending our earlier results in the plane. First of all, we prove that open mappings in W 1,ϕ loc under the Calderon type condition on ϕ have the total differential a.e. that is a generalization of well-known theorems of Gehring-Lehto-Menchoff in the plane and of V¨ ais¨ al¨ a in R n , n  3. Under the same condition on ϕ, it is shown that continuous mappings f in W 1,ϕ loc , in particular, f ∈ W 1,p loc for p>n − 1 have the (N )-property by Lusin on a.e. hyperplane. Our examples show 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 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 loc and, in particular, f ∈ W 1,p 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. 2000 Mathematics Subject Classification: Primary 30C65; Secondary 30C75 Key words: moduli of families of surfaces, Sobolev classes, Orlicz-Sobolev classes, lower Q- homeomorphisms, ring Q-homeomorphisms, mappings of finite distortion, local and boundary behav- ior. Contents 1. Introduction ................................................................................ 2 2. Preliminaries ............................................................................... 6 3. Differentiability of open mappings .......................................................... 9 4. The Lusin and Sard properties on surfaces ................................................. 11 5. Moduli of families of surfaces .............................................................. 15 6. Lower and ring Q-homeomorphisms ....................................................... 18 7. Lower Q-homeomorphisms and Orlicz-Sobolev classes ...................................... 22