Some Boundary Value Problems With Involution for the Nonlocal Poisson Equation

Abstract

In this paper, we study new classes of boundary value problems for a nonlocal analogue of the Poisson equation. The boundary conditions, as well as the nonlocal Poisson operator, are specified using transformation operators with orthogonal matrices. The paper investigates the questions of solvability of analogues of boundary value problems of the Dirichlet and Neumann type. It is proved that, as in the classical case, the analogue of the Dirichlet problem is unconditionally solvable. For it, theorems on the existence and uniqueness of the solution to the problem are proved. An explicit form of the Green's function, a generalized Poisson kernel, and an integral representation of the solution are found. For an analogue of the Neumann problem, an exact solvability condition is found in the form of a connection between integrals of given functions. The Green's function and an integral representation of the solution of the problem under study are also constructed.