Abstract:
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 Green's 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 Green's function, and the integral representation are also found.
The study of the first author was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan within the framework of a scientific project no. AP08855810. The study of the second author was partly supported by Act 211 of the Government of the Russian Federation, contract no. 02.A03.21.0011
Citation:
B. Kh. Turmetov, V. V. Karachik, “On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:4 (2021), 651–667
\Bibitem{TurKar21}
\by B.~Kh.~Turmetov, V.~V.~Karachik
\paper On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2021
\vol 31
\issue 4
\pages 651--667
\mathnet{http://mi.mathnet.ru/vuu793}
\crossref{https://doi.org/10.35634/vm210409}
Linking options:
https://www.mathnet.ru/eng/vuu793
https://www.mathnet.ru/eng/vuu/v31/i4/p651
This publication is cited in the following 3 articles:
B. Kh. Turmetov, “On solvability of some boundary-value problems for the non-local Poisson equation with fractional-order boundary operators”, Probl. anal. Issues Anal., 13(31):3 (2024), 118–134
A. B. Imanbetova, A. A. Sarsenbi, B. N. Seilbekov, “Inverse problems for the beam vibration equation with involution”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 33:3 (2023), 452–466