Salimov R. Non-conformal modulus method in the theory of mappings with finite distortion

In the thesis, the method of the non conformal modulus is developed to research differential, local, asymptotic and boundary properties of mappings with finite distortion, ring and lower Q-homeomorphisms defined in terms of non conformal p-modulus. The characterization of ring and lower Q-homeomorphisms in terms of p-modulus is obtained and the relationship between these homeomorphisms is established. Analogues of the M.O. Lavrentyev inequality on the distortion of the area of a disk for quasiconformal mappings, the Gehring lemma on local Lipschitz property and the Ikoma--Schwartz theorem are proved for ring and lower Q-homeomorphisms defined in terms of p-modulus. It is established that homeomorphic solutions of degenerate Beltrami equations with generalized derivatives are ring and lower Q-homeomorphisms, where Q is a tangent dilatation, and generalized theorems on continuous and homeomorphic extension of these solutions and their asymptotic behavior at infinity are proved. The general conditions on the tangential dilatation sufficient for the existence of regular solutions of the Dirichlet problem for the degenerate Beltrami equation in an arbitrary Jordan domain are established. The connection of Sobolev classes in the domains of complex plane, as well as the classes of Orlicz-Sobolev in the space Rn under the Calderon type condition, with lower and ring Q-homeomorphisms defined in terms of p-modulus is established. Sufficient conditions of local and logarithmic Holder properties, power and logarithmic orders of growth of homeomorphisms belonging to the mentioned Sobolev and Orlicz--Sobolev classes are found. Sufficient conditions for Q-mappings defined in terms of p-modulus to belong to the Sobolev classes are established, and a generalization of the Bojarski-Iwaniec result on nonvanishing Jacobian is proved. Upper bounds of the Jacobian and the operator norm of the Jacobian matrix, inner and outer dilatations of ring Q-mappings with respect to p-modulus are obtained. An analogue of the Vaisala's result on absolute continuity on lines for mappings satisfying p-modulus inequality with respect to spatial cylinders is also given.