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Choquet simplices and Harnack inequalities. (English) Zbl 0639.46012
Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 339-343 (1987).
[For the entire collection see Zbl 0627.00012.]
Let S be a metrizable Choquet simplex and E the set of all extreme points of S. Given \(x\in S\), let \(\mu_ x\) be the unique maximal measure representing the point x and let face(x) be the smallest face of S containing x. By the solution of the Dirichlet problem for a boundary function f on E we understand the affine function \(u_ f\) defined on \(D_ f=\{x\in S:f\in L\) \(1(\mu_ x)\}\) (which is always a face of S) by \(u_ f(x)=\mu_ x(f).\)
The aim of this paper is to describe the points \(x\in S\) for which an analogue of the Harnack inequality is satisfied on face(x), i.e., for any compact set \(K\subset face(x)\) there is a number \(\alpha_ K\) such that for every continuous affine function f:face(x)\(\to [0,\infty)\) we have \(\sup_{y\in K}f(y)\leq \alpha_ Kf(x)\). The author proves that this is the case iff the restriction to face(x) of the solution of the Dirichlet problem is continuous for every boundary function from L \(1(\mu_ x)\).
Reviewer: I.Raşa
46A55 Convex sets in topological linear spaces; Choquet theory
Zbl 0627.00012
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