Compatibility conditions for Dirichlet and Neumann problems of Poisson's equation on a rectangle

It is a well-known fact that the solution of Poisson's equation on a rectangle lacks regularity. Even for a smooth inhomogeneity, corner singularities arise in the derivatives of the solution. The very form of these singularities is of particular interest in numerical analysis; more precisely for the analysis of dimension splitting methods applied to parabolic equations. In this work, necessary and sufficient conditions on the inhomogeneity are derived which ensure a higher regularity of the solution of the Dirichlet or the Neumann problem - the so called compatibility conditions.