Dirichlet problem and subclasses of Baire-one functions

Abstract

Let U be a bounded open subset of ℝd, d ≥ 2 and f ∈ C(∂U). The Dirichlet solution fCU of the Dirichlet problem associated with the Laplace equation with a boundary condition f is not continuous on the closure Ū of U in general if U is not regular but it is always Baire-one. Let H(U) be the space of all functions continuous on the closure Ū and harmonic on U and F(H(U)) be the space of uniformly bounded absolutely convergent series of functions in H(U). We prove that fCU can be obtained as a uniform limit of a sequence of functions in F(H(U)). Thus fCU belongs to the subclass B1/2 of Baire-one functions studied for example in [8]. This is not only an improvement of the result obtained in [10] but it also shows that the Dirichlet solution on the closure Ū can share better properties than to be only a Baire-one function. Moreover, our proof is more elementary than that in [10]. A generalization to the abstract context of simplicial function space on a metrizable compact space is provided. We conclude the paper with a brief discussion on the solvability of the abstract Dirichlet problem with a boundary condition belonging to the space of differences of bounded semicontinuous functions complementing the results obtained in [17].