Solvability of one nonlocal Dirichlet problem for the Poisson equation

Valery Karachik, Batirkhan Turmetov

In this paper the solvability of a new class of nonlocal boundary value problems for the Poisson equation is studied. These problems are a generalization of the classical Dirichlet boundary value problem. Existence and uniqueness theorems for the considering problem are proved. An integral representation of the solution is established. The notion of the Green's function for the problem under consideration is introduced and an explicit form of this function is constructed. The corresponding spectral issues are also studied, namely eigenfunctions and eigenvalues of the considered problem are found. For one particular case of the problem the completeness of the system of eigenfunctions is proved.