On solvability of some boundary value problems for a nonlocal Poisson equation with periodic conditions

О РАЗРЕШИМОСТИ НЕКОТОРЫХ КРАЕВЫХ ЗАДАЧ ДЛЯ НЕЛОКАЛЬНОГО УРАВНЕНИЯ ПУАССОНА С ПЕРИОДИЧЕСКИМИ УСЛОВИЯМИ
Turmetov B.K.
2025 Udmurt State University

Vestnik Udmurtskogo Universiteta: Matematika, Mekhanika, Kompyuternye Nauki
2025 #35 Issue 1 137 - 154 pp.

In the present paper, a nonlocal analog of the Laplace operator is introduced by means of involution-type mappings. New classes of boundary value problems are studied for the corresponding nonlocal analog of the Poisson equation in a unit sphere. In the problems under consideration, the boundary conditions are given in the form of a relation between the value of the unknown function in the upper hemisphere and the value in the lower hemisphere. The problems under study generalize the known periodic and antiperiodic boundary value problems for circular regions. The problems are solved by reducing them to two auxiliary problems with Dirichlet and Neumann boundary conditions for the nonlocal analog of the Poisson equation. Using known statements for the obtained auxiliary problems, we prove theorems on the existence and uniqueness of solutions of the main problems. Exact conditions for the solvability of the investigated problems are found, and integral representations of the solutions are obtained. Spectral issues related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found. The theorems on completeness of the system of eigenfunctions in the space L₂ are proved.

Dirichlet problem , eigenfunctions , eigenvalues , involution , Neumann problem , periodic conditions , Poisson equation

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Department of Mathematics, Khoja Akhmet Yassawi International Kazakh-Turkish University, ul. Bekzata Sattarkhanova, 29, Turkistan, 161200, Kazakhstan
Alfraganus University, ul. Yukori Karakamysh, 2 a, Tashkent, 100190, Uzbekistan

Department of Mathematics
Alfraganus University

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