On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution
О РАЗРЕШИМОСТИ КРАЕВЫХ ЗАДАЧ ДИРИХЛЕ И НЕЙМАНА ДЛЯ УРАВНЕНИЯ ПУАССОНА С МНОЖЕСТВЕННОЙ ИНВОЛЮЦИЕЙ
Turmetov B.K. Karachik V.V.
2021Udmurt State University
Vestnik Udmurtskogo Universiteta: Matematika, Mekhanika, Kompyuternye Nauki
2021#31Issue 4651 - 667 pp.
Transformations of the involution type are considered in the space Rl, l > 2. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Greens function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Greens function, and the integral representation are also found.
Dirichlet problem , Multiple involution , Neumann problem , Nonlocal laplace operator , Poisson equation , Transformation matrix
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Department of Mathematics, Khoja Akhmet Yassawi International Kazakh-Turkish University, ul. B. Sattarkhanov, 29, Turkistan, 161200, Kazakhstan
Department of Mathematical Analysis and Methods of Mathematics, South Ural State University, pr. Lenina, 76, Chelyabinsk, 454080, Russian Federation
Department of Mathematics
Department of Mathematical Analysis and Methods of Mathematics
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