Dovhopiaty O. On the theory of local and boundary behavior of plane and space mappings

The thesis is devoted to the study of the properties of mappings with finite distortion, which have been actively studied during the last 25-30 years, as well as the Beltrami equation and the Dirichlet problem. The first section is devoted to the theory of boundary behavior of mappings in the Euclidean space. In addition, modular inequalities of the Poletsky type have been obtained. The section consists of three subsections. In the first subsection, the continuous extension of the mappings with the inverse Poletsky inequality to the boundary is obtained. The result was proved under the conditions that the majorant in this inequality is integrable, the definition domain has a weakly flat boundary, and the mapped domain is locally connected at its boundary. In the second subsection, we have obtained the upper inverse modulus condition of the Poletsky type in which some analogue of the inner dilatation is used. The third subsection is devoted to the Hӧlder logarithmic continuity of mappings with the inverse Poletsky inequality at the boundary points in the case when the majorant in this inequality is integrable, and the mapped domain is bounded and convex. The second section is devoted to compactness theorems of classes of solutions of the Beltrami equation and the Dirichlet problem. The section consists of three subsections. In the first subsection, the author has proved theorems about compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation, the characteristics of which have a compact support and satisfy certain integral constraints. The second subsection is devoted to proving theorems about compact classes of homeomorphisms with hydrodynamic normalization, which are solutions of the Beltrami equation, the characteristics of which have a compact support and satisfy certain restrictions of the set-theoretic type. The third subsection is devoted to the problem of the compactness of solutions of the Dirichlet problem for the Beltrami equation in some simply connected domain. The third section is devoted to the existence of solutions to the Beltrami equation and contains three subsections. The first subsection concerns the existence of solutions of quasilinear Beltrami equations with two characteristics. The second subsection is devoted to the existence of solutions of quasilinear Beltrami equations with two characteristics and hydrodynamic normalization. The last (third) subsection concerns spatial mappings with an analogue of hydrodynamic normalization. The author of the manuscript proved that homeomorphisms with the specified property form equicontinuous families under certain conditions of their complex characteristic.