Algebraic Aspects of the Dirichlet Problem

Abstract

(1.3) Kn(x) = ωn 1− ||x|| ||y − x||n and ωn is the reciprocal of the surface area of the sphere Sn−1. Many classical analytic properties of the solution u (such as e.g. regularity up to the boundary) can be gleaned from this formula. In this paper, we shall address some questions of algebraic nature which do not seem to follow as easily from (1.2). We take as our starting point the following well known result: If f(x) is a polynomial of degree m, then the solution u(x) to the Dirichlet problem (1.1) is a polynomial of degree at most m. The precise origin of this result is difficult to trace (see e.g. [F] for the result in R), but a simple and elementary proof based on linear algebra can be found in e.g. [K], [KS], and [Sh] (where the conclusion is in fact proved for the Dirichlet problem in an arbitary ellipsoid). In light of this basic result, it seems natural to ask if the solution operator preserves wider classes of algebraic functions: Question A. Assume that the data f(x) is a rational function without poles on Sn−1. Is then the solution u(x) of the Dirichlet problem (1.1) necessarily rational?