Complementability of Spaces of Harmonic Functions

Let U be a bounded open set of the Euclidean space R-d and let H(U) denote the space of all real-valued continuous functions on (U) over bar that are harmonic on U. We present a sufficient condition on the set partial derivative U-reg of all regular points of U that ensures that H(U) is complemented in C((U) over bar). We also present examples showing that this condition is not necessary. The proof of the positive result is based upon a general result on complementability of a simplicial function space in a C(K) space.