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Choquet’s theory and the Dirichlet problem. (English) Zbl 1046.31001
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).

MSC:
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31C35 Martin boundary theory
46A55 Convex sets in topological linear spaces; Choquet theory
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