Dirichlet problem and subclasses of Baire-one functions
Let \(U\subset\mathbb{R}^d\), \(d\geq 2\), be a bounded open set and \(f\in C(\partial U)\) a boundary condition. The author proves that the classical Dirichlet solution for \(f\) belongs to the space \(B_{1/2}(H(U))\), a subspace of Baire-one functions on \(\overline{U}\) (recall that \(B_{1/2}(H(U))\) are by definition those functions which are uniform limits of sequences of absolutely convergent series of harmonic functions) and that \(B_{1/2}(H(U))\subsetneq B_1^{bb}(H(U))\), which means that the above mentioned result is a strict generalization of what was proved by \textit{J. Lukeš} et al. [Isr. J. Math. 134, 255--287 (2003; Zbl 1031.35011)]. In the second part of the paper, some analogous results are proved in the abstract context of Choquet theory on function spaces. Finally, an abstract Dirichlet problem for the boundary condition belonging to the class of differences of semicontinuous functions is discussed.
scientific article; zbMATH DE number 6932210
scientific article