On an Samarskii-Ionkin boundary value problem for the Poisson equation in a disk

In this paper we consider a Samarskii-Ionkin type boundary value problem for the Poisson equation in a disk and prove its well-posedness. The possibility of separation of variables is justified. We construct an explicit form of the Green function for this problem and obtain an integral representation of the solution. We consider spectral properties and prove the completeness of eigenfunctions. In addition, we note that unlike the one-dimensional case the system of root functions of the problems consists only of eigenfunctions.