Geodesic
On a nonlocal boundary value problem with an oblique derivative
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 81-93

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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.
Mots-clés : oblique derivative, Laplace equation, orthogonal matrix, existence of solution
Keywords: nonlocal problem, Helder class, smoothness of solution, uniqueness of solution. 
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     author = {K. Zh. Nazarova and B. Kh. Turmetov and K. I. Usmanov},
     title = {On a nonlocal boundary value problem with an oblique derivative},
     journal = {\v{Z}urnal Srednevol\v{z}skogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {81--93},
     publisher = {mathdoc},
     volume = {22},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a5/}
}
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K. Zh. Nazarova; B. Kh. Turmetov; K. I. Usmanov. On a nonlocal boundary value problem with an oblique derivative. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 22 (2020) no. 1, pp. 81-93. http://geodesic.mathdoc.fr/item/SVMO_2020_22_1_a5/