Dirichlet problem and subclasses of Baire-one functions
type: Item
instance of: scholarly article
title: Dirichlet problem and subclasses of Baire-one functions
zbMATH Open document ID: 1409.46021
author: Petr Pošta
published in: Israel Journal of Mathematics
publication date: 2018-09-05T00:00:00Z
review text: 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.
reviewed by: Marek Cúth
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 31D05
zbMATH DE Number: 6932210
zbMATH Keywords: Dirichlet solution
zbMATH Keywords: Baire-one function
zbMATH Keywords: Choquet theory
MaRDI profile type: Publication
full work available at URL: s11856-018-1707-z
OpenAlex ID: W2807427125
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DOI: 10.1007/S11856-018-1707-Z
description: scientific article; zbMATH DE number 6932210