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This article is cited in 1 scientific paper (total in 1 paper)
Mathematics
On a nonlocal boundary value problem with an oblique derivative
K. Zh. Nazarova, B. Kh. Turmetov, K. I. Usmanov Kh. Yasavi International Kazakh-Turkish University
Abstract:
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.
Keywords:
oblique derivative, nonlocal problem, Laplace equation, orthogonal matrix, Helder class, smoothness of solution, existence of solution, uniqueness of solution.
Citation:
K. Zh. Nazarova, B. Kh. Turmetov, K. I. Usmanov, “On a nonlocal boundary value problem with an oblique derivative”, Zhurnal SVMO, 22:1 (2020), 81–93
Linking options:
https://www.mathnet.ru/eng/svmo762 https://www.mathnet.ru/eng/svmo/v22/i1/p81
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Abstract page: | 140 | Full-text PDF : | 39 | References: | 17 |
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