We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz-Sobolev classes W 1,ϕ loc under a condition of the Calderon type for the function ϕ and, in particular, in the Sobolev classes W 1,p loc for p > n − 1.

It is shown that, under a Calderón type condition on the function ϕ, the continuous open mappings that belong to the Orlicz-Sobolev classes W 1,ϕ loc have total differential almost everywhere; this generalizes the well-known theorems of Gehring-Lehto-Menchoff in the case of R 2 and of Väisälä in R n , n ≥ 3. Appropriate examples show that the Calderón type condition is not only sufficient but also necessary. Moreover, under the same condition on ϕ, it is also proved that the continuous mappings of class W 1,ϕ loc and, in particular, of class W 1,p 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 ϕ, the homeomorphisms f with finite distortion of class W 1,ϕ loc and, in particular, those belonging to W 1,p 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 n−1 f . 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. Part 1. Differentiability and behavior of Hausdorff 's measures in the Orlicz-Sobolev classes2010 Mathematics Subject Classification. Primary 46E35. Key words and phrases. Moduli of families of curves and surfaces, mappings with bounded and finite distortion, differentiability, Lusin and Sard properties, Sobolev classes, Orlicz-Sobolev classes, boundary and local behavior.

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It is shown that a homeomorphic solution of the Beltrami equation ∂f = μ ∂f in the Sobolev class W 1,1 loc is a so-called ring and, simultaneously, loweris the dilatation ratio of this equation. On this basis, the theory of the boundary behavior of such solutions is developed and, under certain conditions on K μ (z), the existence of regular solutions is established for the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. Also, the existence of pseudoregular as well as many-valued solutions is proved in the case of arbitrary finitely connected domains bounded by mutually disjoint Jordan curves.