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New methods of analysis of boundary-value problems in function spaces

Work number - M 7 AWARDED

Authors: Anop A. V., Soldatov V. O., Chepurukhina I. S.

The work is presented by Institute of Mathematics of NAS of Ukraine.

The work is devoted to the development of new and improvement of known methods for investigation of elliptic boundary-value problems and one-dimensional boundary-value problems. The main attention is paid to the application of methods of functional analysis such as scale of functional spaces, interpolation of functional spaces, and the theory of Fredholm operators. The work consists of three parts and presents the results of its authors obtained in the department of nonlinear analysis of Institute of Mathematics of the NAS of Ukraine within 2013 – 2017. It has a theoretical character.

The work presents the theory of solvability of general elliptic boundary-value problems in Hörmander spaces which form the extended Sobolev scale. This scale consists of all Hilbert spaces that are interpolating ones between inner-product Sobolev spaces. The core of this theory is formed by theorems on the Fredholm property of these problems and on induced isomorphisms on corresponding pairs of Hörmander spaces and also by theorems on a priori estimates of the solutions to these problems and the local regularity of the solutions in the Hörmander spaces. Regular and nonregular elliptic problems, formally mixed elliptic problems, parameter-elliptic problems, and elliptic problems for systems of differential equations are investigated.

The theory of solvability of B. Lawruk elliptic boundary-value problems on the scales of Hörmander spaces is built. Unlike the classical boundary value problems, Lawruk elliptic problems contain additional unknown functions in boundary conditions. Theorems on the Fredholm property of these problems in the Hörmander spaces and their modifications in the sense of Ya. A. Roitberg and a priori estimates and the local regularity of the solutions to the problems are proved. A part of the results of this theory is new for the Sobolev spaces. As an application of Hörmander spaces, the authors obtain new finer sufficient conditions for the continuity of givengeneralized partial derivatives of solutions to various elliptic boundary-value problems.

The work introduce and investigate the maximal broad classes of linear boundary-value problems for systems of ordinary differential equations whose solutions goes through the normed spaces of continuously differentiable functions or the Hölder–Zygmund spaces. For the parameter dependent problems introduced, constructive criteria for the solutions to be continuous with respect to the parameter in these spaces are established. It is shown that the error and discrepancy of the solutions are of the same degree. These results are applied to the study of multi-point boundary problems.

The results of the work and methods for their obtaining are used in the spectral theory of elliptic differential operators, in the theory of parabolic problems, in the spectral theory of boundary-value problems of mathematical physics.

The work consists of 53 scientific works, among which 25 articles in the lead national and international professional editions. Total number of references to articles and the corresponding h-index: Google Scholar – 103citations and h-index = 7; SCOPUS – 19 citations and h-index = 3; Web of Science - 30 citatins and h-index = 4; MathSciNet – 40 citations and h-index = 4.