Chepurukhina I. Lawruk elliptic boundary-value problems in Hermander spaces

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, 2015. In the thesis, we have built the theory of solvability of Lawruk elliptic boundary-value problems in the classes of inner product Hermander spaces; namely, in the extended Sobolev scale and the refined Sobolev scale. In contrast to classical elliptic boundary-value problems, Lawruk elliptic problems contain additional unknown functions in boundary conditions. The extended Sobolev scale consists of all Hermander inner product spaces for which the index of regularity of distributions is a function parameter RO-varying at infinity in the sense of Avakumovich. Its important subclass---the refined Sobolev scale---is attached to the Hilbert-Sobolev scale by the number parameter. These scales of Hermander spaces are calibrated more finely by means of the function parameter than this Sobolev scale. Their application allows us to obtain more precise results on properties of elliptic problems than this is possible in the framework of the theory of Sobolev spaces. In the thesis, we prove theorems on the Fredholm property of Lawruk elliptic boundary-value problems in appropriate pairs of Hermander spaces that belong to the refined Sobolev scale and consist of regular distributions. We prove theorems about the Fredholm property of these problems on the complete refined Sobolev scale modified in the sense of Roitberg. We also prove a Lions-Magenes-type theorem on the Fredholm property of these problems in Hermander and Sobolev spaces that contain irregular distributions of arbitrary negative order. A priori estimates are established for generalized solutions to Lawruk elliptic boundary value problems considered in Hermander spaces and their modifications. We prove theorems about the regularity of the solutions in these spaces. We find new sufficient conditions under which generalized derivatives (of a given order) of these solutions are continuous; specifically, we obtain sufficient conditions for the generalized solutions to be continuous.