ISSN 2079-6900 (Print)
ISSN 2587-7496 (Online)

MSC2020 35J25

On a nonlocal boundary value problem with an oblique derivative

K. Zh. Nazarova1, B. Kh. Turmetov2, K. I. Usmanov3

Annotation The work studies the solvability of a nonlocal boundary value problem for the Laplace equation. The nonlocal condition is introduced using transformations in the $R^{n}$ space carried out by some orthogonal matrices. Examples and properties of such matrices are given. To study the main problem, an auxiliary nonlocal Dirichlet-type problem for the Laplace equation is first solved. This problem is reduced to a vector equation whose elements are the solutions of the classical Dirichlet probem. Under certain conditions for the boundary condition coefficients, theorems on uniqueness and existence of a solution to a problem of Dirichlet type are proved. For this solution an integral representation is also obtained, which is a generalization of the classical Poisson integral. Further, the main problem is reduced to solving a non-local Dirichlet-type problem. Theorems on existence and uniqueness of a solution to the problem under consideration are proved. Using well-known statements about solutions of a boundary value problem with an oblique derivative for the classical Laplace equation, exact orders of smoothness of a problem's solution are found. Examples are also given of the cases where the theorem conditions are not fulfilled. In these cases the solution is not unique. oblique derivative, nonlocal problem, Laplace equation, orthogonal matrix, Helder class, smoothness of solution, existence of solution, uniqueness of solution

1Kulzina Zh. Nazarova, Associate Professor of Mathematics, Akhmet Yassawi University (29 Sattarkhanov Av., Turkistan 161200, Kazakhstan), Ph. D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-2093-1879, gjnazarova@mail.ru

2Batirkhan Kh. Turmetov, Professor of Mathematics, Akhmet Yassawi University (29 Sattarkhanov Av., Turkistan 161200, Kazakhstan), Dr. Sci. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-2093-1879, turmetovbh@mail.ru

3Kairat Id. Usmanov, Associate Professor of Mathematics, Akhmet Yassawi University (29 Sattarkhanov Av., Turkistan 161200, Kazakhstan), Ph. D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-2093-1879, y_kairat@mail.ru

Citation: K. Zh. Nazarova, B. Kh. Turmetov, K. I. Usmanov, "[On a nonlocal boundary value problem with an oblique derivative]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,22:1 (2020) 81–93 (In Russian)

DOI 10.15507/2079-6900.22.202001.81-93