On Existence Theorems of Potential Theory and Conformal Mapping

Abstract

The object of the present paper is to give a new set of proofs of some fundamental existence theorems of conformal mapping and potential theory by using the notion of a kernel function introduced by Bergman [2, 3, 4]. The proofs which we present here are a more or less natural outcome of the recent researches done in this theory [5, 6, 7, 11, 12]. Our new attack upon the existence problem has on the one hand the advantage of generality, since it is applicable in the case of partial differential equations of elliptic type and since for functions of several complex variables the theory of the kernel function is well developed, while on the other hand the approach is simple and elementary in comparison with, say, the method using the Dirichlet principle. For potential theory and conformal mapping in multiply-connected plane domains, our proof is probably as simple as any known, and even for simply-connected domains the method has the advantage that it is based on a general theory rather than upon special artifices. A further point which we gain is that the kernel functions associated with the various norms of function theory and elliptic partial differential equations are well adapted to numerical computation, in view of their relation to orthogonal functions, so that we obtain at one and the same time an existence proof and a computational algorithm. The most serious drawback in our method is, perhaps, that we must make assumptions upon the smoothness of the boundary of the domains we consider, so that the general case is reached only after a topological approximation argument is given. From the broader point of view, our treatment is important in that it yields a unified attack upon various existence problems. An extensive class of existence theorems of function theory of one or more variables and of partial differential equations of elliptic type can be developed using one basic procedure. This procedure can be outlined as follows. We set up the reproducing kernel function in a given function space by solving a suitable extremal problem, or alternatively by means of orthogonal functions, and we investigate the scalar product of the kernel function with what may be described as the fundamental singularity associated with this space. A local argument is used to study the behavior of this scalar product as the infinity of the fundamental singularity crosses the boundary of the domain of definition of the functions in our class, and the desired existence theorems follow from the geometric properties thus obtained. The local investigations depend in a general way upon a knowledge of our problem in simple domains, such as, for example, the unit circle.