Choquet's theory and the Dirichlet problem.
type: Item
instance of: scholarly article
title: Choquet's theory and the Dirichlet problem.
zbMATH Open document ID: 1046.31001
DOI: 10.1016/S0723-0869(02)80021-4
author: J. Lukeš
author: Ivan Netuka
author: Jiří Veselý
published in: Expositiones Mathematicae
publication date: 2002-01-01T00:00:00Z
review text: This nice survey illustrates the importance of Choquet's theory for many areas of mathematics. In a first section various results are presented which are special cases of the fundamental theorem on integral representation (results on convex sets in Euclidean space, convex functions, doubly stochastic matrices, typically real holomorphic functions, completely monotone functions, solutions to the Helmholtz equation, Fourier transform of measures, harmonic functions on a ball, invariant and ergodic measures). The next section discusses the theorem on integral representation as a reformulation of Krein-Milman's theorem. The heart of the paper is Choquet's theorem. Some definitions are needed: Let \(\mathcal H\) be a vector space of continuous real functions on a compact metric space \(K\) such that \(1\in\mathcal H\) and the points of \(K\) are separated by \(\mathcal H\). A Borel measure \(\mu\) on \(K\) is an \(\mathcal H\)-representing measure \(\mu\) for a point \(x\in K\) if \(h(x)=\int h\,d\mu\) for every \(h\in\mathcal H\); the Choquet boundary \(\text{Ch}_{\mathcal H}K\) is the set of all points \(x\in K\) such that the Dirac measure at \(x\) is the only \(\mathcal H\)-representing measure for \(x\). Now Choquet's theorem states that, for every \(x\in K\), there exists a representing measure which is concentrated on \(\text{Ch}_{\mathcal H}K\). An important consequence is an abstract maximum principle involving \(\mathcal H\)-convex functions and the Choquet boundary. Having discussed further general material (\(\mathcal H\)-exposed points, \(\mathcal H\)-affine functions, simplicial spaces etc.) the authors concentrate on questions related to the Dirichlet problem in potential theory (classical and generalized Dirichlet problem, the spaces \(H(U)\), Keldysh operators, Martin representation, restricted mean value property).
reviewed by: Wolfhard Hansen
Mathematics Subject Classification ID: 31B10
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 46A55
zbMATH DE Number: 1810506
zbMATH Keywords: Choquet boundary
zbMATH Keywords: representing measure
zbMATH Keywords: maximum principle
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: harmonic function
Wikidata QID: Q116636124
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1810506