On Baire and Harmonic Functions

We consider two spaces of harmonic functions. First, the space H(U) of functions harmonic on a bounded open subset U of R and continuous to the boundary. Second, the space H0(K) of functions on a compact subset K of R n which can be harmonically extended on some open neighbourhood of K. A bounded open subset U of R is called stable if the space H(U) is equal to the uniform closure of H0(U ). We will discuss whether the stability of U is a necessary condition for the equality of systems of functions which are pointwise limits of the spaces H(U) and H0(U). Introduction Let U be a bounded open subset of R and f be a continuous function on its boundary ∂U . We define a space H(U) of functions harmonic on a bounded open subset U of R and continuous to the boundary. The Dirichlet problem for U and f is to find a function h ∈ H(U), that is, harmonic on U and continuous to the boundary, such that h = f on ∂U . It is well known that the solution to the Dirichlet problem does not need to exist for all bounded open sets and boundary conditions. However, it has been proved recently by Lukeš et al. (2003) that the generalized solution of the Dirichlet problem for a continuous boundary condition can be achieved as a pointwise limit of a sequence of functions in H(U) and, moreover, this sequence can be chosen bounded. In the same article they posed a question about the structure of the space of pointwise limits of functions in H(U) (see Example 3.12. there). To the best of our knowledge, this problem is still open. A similar question for the space of pointwise limits of functions in H0(K) where K is a compact subset of R was answered by Gardiner and Gustafsson (2005). The motivation for our article is this: it is obvious that the stability of the set U is a sufficient condition for both systems of pointwise limits being equal to each other. We ask whether the stability of the set U is a necessary condition as well. Definitions and preliminary results Throughout this article, U will denote a bounded open subset of R and K a compact subset of R. We also assume that n ≥ 2 throughout this article if not said otherwise. For reader’s convenience, we repeat once more that the space H(U) is defined by H(U) = {f : U → R, f is harmonic on U and continuous on U} and the space H0(K) by H0(K) = {f : K → R, f can be harmonically extended on some open neighbourhood of K}. Obviously, both of these spaces can be viewed as subspaces of the space of continuous function on a compact set U or K, respectively. Both of them contain constant functions and separate points. The first one is closed under uniform convergence, the latter one need not be. A bounded open subset U of R is called stable if the space H(U) is equal to the uniform closure of H0(U). 84 WDS'09 Proceedings of Contributed Papers, Part I, 84–88, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS