A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory (Q1434541)

The paper contains an interesting account and also a proof of D. A. Grave's method for solving classical plane boundary value problems for Green's function of the Laplace equation in regions whose boundaries are smooth analytic curves defined by finite-order polynomials. This method has certain advantages when compared with the method of constructing Green's function by conformally mapping the original region onto the unit disk. A class of regions for which Grave's method gives an explicit analytic solution in convergent-series form is identified (Grave's method is a generalization of the conformal mapping method for simplest regions). The method is illustrated by constructing Green's function for Dirichlet problems for the interior of a parabola and an ellipse.