On the application of the regularization method to the construction of a classical solution of Poisson's equation

Necessary and sufficient conditions are found for the existence of a classical solution of Poisson's equation $\Delta u=f$ with continuous function $f$ in a bounded planar domain. By virtue of the known smoothness properties of a generalized harmonic function, these conditions also ensure that all generalized solutions of Poisson's equation are classical in this domain. Particular classes of functions $f$ satisfying the conditions of existence of a classical solution are described.