**MSC2020** 35J25

### On a nonlocal boundary value problem with an oblique derivative

#### K. Zh. Nazarova^{1}, B. Kh. Turmetov^{2}, K. I. Usmanov^{3}

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. |
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Keywords | oblique derivative, nonlocal problem, Laplace equation, orthogonal matrix, Helder class, smoothness of solution, existence of solution, uniqueness of solution |

^{1}**Kulzina 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^{2}**Batirkhan 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^{3}**Kairat 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