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Bayzaev Sattor	ON MANIFOLDS OF SOLUTIONS OF MODERATE GROWTH OF AN ELLIPTIC SYSTEM OF FOUR EQUATIONS AND EQUATIONS WITH THE BITSADZE OPERATOR	36

Authors: Bayzaev Sattor – Doctor of Physical and Mathematical Sciences, Professor of the Department of Mathematical Disciplines and Modern Natural Science, Tajik State University of Law, Business and Politics (Tajikistan Republic, Khujand), Vositova Dilorom Abdurasulovna – Candidate of Physical and Mathematical Sciences, Dotsent of the Department of Mathematical Analysis named after professor A.Mukhsinov, State Educational Instituon “KhSU named after academician  B.Gafurov” (Tajikistan Republic, Khujand)

JOURNAL NUMBER: 3(62). YEAR OF ISSUE: 2022. LANGUAGE OF THE ARTICLE: Russian

ANNOTATION

The article considers a real elliptic system of four partial differential equations with two first-order independent variables

                                                                                       (1)

where is a second-order identity matrix, a real constant matrix of the fourth order, and an equation with constant complex coefficients of the form

                                                                                                                 (2)

Equation (2) is a generalization of the system of Bitsadze equations for which the Dirichlet problem is not Noetherian. It turned out that for systems of the form (1) and equations of the form (2), the problem of solutions bounded on the entire plane may not be Noetherian. The article gives examples of systems of types (1) and (2) that have an infinite number of linearly independent solutions bounded on the entire plane. This fact indicates that the study of systems of the form (1) and (2) in spaces of functions defined on the whole plane is relevant.

For systems (1) and (2), we study problems of solutions defined in the whole plane and belonging to the space of distributions of moderate growth, in particular, to the space of functions growing at infinity no faster than   Schemes for finding solutions to these systems from the corresponding spaces are proposed. The structures of supports of the Fourier image of solutions, which can be point like or consist of continuous closed curves, are described. For equation (9), it is established that the space of solutions of moderate growth for any coefficients  is nonzero, and the space of solutions growing at infinity  no faster than at  is infinite-dimensional, and at  - finite-dimensional.

KEY WORDS

partial differential equations, elliptic system, bianalytic functions, metaanalytic functions, solutions of polynomial growth, Schwartz spaces.

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