Abstract
The purpose of this paper is to treat a generalization of the elementary potential theory of the unit circle. The main tools are those of the theory of transformation groups. For this reason, the domains treated are required to have a large amount of symmetry; they form a class of homogeneous spaces first discussed in full generality by E. Cartan [6]. A certain class of partial differential equations is shown to arise naturally from the requirements of symmetry. These equations are elliptic on the interior of the domain but degenerate on the boundary; very few existence theorems or explicit solutions for such equations are known. Fortunately, the requirements of symmetry also give simple and explicit solutions by a generalization of the Poisson integral formula.
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