On an analog of periodic boundary value problems for the Poisson equation in the disk

We consider a periodic and an antiperiodic problem for the Poisson equation in the unit disk and prove their well-posedness. The possibility of separation of variables is justified. We construct the Green functions of these problems in closed form and obtain an integral representation of the solution. The problems are self-adjoint, and we construct all eigenvalues and eigenfunctions in closed form.