Singularities encountered by the analytic continuation of solutions to dirichlet's problem

Abstract

We study the analytic continuation of solutions to Dirichlet's problem with real entire data. The domains we study are quadrature domains, domains bounded by ellipses and domains which are bounded by, loosely speakingk:th roots of ellipses. We give a new proof of the known result (see[1]) that when the domain is bounded by an ellipse the solution extends as a harmonic function to all of R 2. Also, we prove the following new results: (1) if the domain is a quadrature domain the solution extends as a multivalued harmonic function to R 2 minus a finite set of points; (2) if the domain is bounded by the k:th root of an ellipse then the solution extends as a multivalued harmonic function to R 2 minus a discrete, but infinite, set of points. In both cases the set of possible singular points is completely determined by the domain and the singularities encountered at these points are algebraic. We also show that these results are essentially sharp, in the sense that the solution with data x 2 + y 2 develops singu...