Anop A. Elliptic boundary-value problems in spaces of generalized smoothness.

The thesis for the scientific degree of the candidate of physical and mathematical sciences by speciality 01.01.02 --- differential equations. --- Institute of mathematics, National Academy of Sciences of Ukraine, Kyiv, 2014. In the theses, properties of general boundary-value elliptic problems are investigated in Hilbert spaces of generalized smoothness that form the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolation spaces for pairs of inner product Sobolev spaces and admits a description in terms of H?rmander function spaces. On the extended Sobolev scale, the character of solvability of elliptic boundary-value problems [both regular and nonregular] and properties of their solutions are investigated. We prove the theorems on the Fredholm property of the operators corresponding to these problems, and the theorems on a priori estimates for their solutions and on the local regularity of the solutions in H?rmander spaces. As an application of this scale, we find new sufficient conditions under which generalized derivatives [of prescribed order] of solutions are continuous. Specifically, new conditions for generalized solutions to be classical are established. We also prove the theorem on the Fredholm property for the operators that correspond to formally mixed elliptic boundary-value problems on the extended Sobolev scale over a multiply connected Euclidean domain. For parameter-elliptic boundary-value problems, we prove that the corresponding operators are isomorphisms on the extended Sobolev scale provided that the absolute value of the complex parameter is large enough. New a priori estimates are established for the solutions to these problems. For elliptic boundary-value problems for systems of partial differential equations, we prove the theorem on the Fredholm property of the corresponding operators on the extended Sobolev scale and the theorems on a priori estimates for solutions and their local regularity.