On the Existence of Green's Function

In this note we shall present a very short proof of the existence of Green's function for Laplace's equation for any domain with sufficiently smooth boundary in any number of independent variables. The proof is based on the continuous dependence of solutions of Laplace's equation on their boundary values. It is a modification of a proof given by Paul Garabedian, see [l]; the difference between the two approaches is that whereas Garabedian operates with a representation of harmonic functions in terms of their boundary data which he obtains by a variational argument, in our argument only the linear and bounded dependence of the solution on the boundary values figures. 1. In this section we shall treat the somewhat simpler two-dimensional case. We consider a bounded domain D whose boundary C consists of a finite number of smooth curves (i.e., curves with continuous tangents) . B is the Banach space of all continuous functions defined on C, normed by the maximum norm. B' is the submanifold of those elements of B for which the boundary value problem can be solved.1