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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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##### References:
 [1] Alfsen, E.M., Compact convex sets and boundary integrals, (1971), Springer-Verlag, MR 56 #3615 · Zbl 0209.42601 [2] Armitage, D.H., The Riesz-Herglotz representation for positive harmonic functions via Choquet’s theorem, (), 229-232, MR 97f:31006 · Zbl 0856.31003 [3] Armitage, D.H.; Gardiner, S.J., Classical potential theory, (2001), Springer-Verlag London, Ltd. Berlin, MR 2001m:31001 · Zbl 0972.31001 [4] Bauer, H., Approximation and abstract boundaries, Amer. math. monthly, 85, 632-647, (1978), MR 80f:41014 · Zbl 0416.41023 [5] Bauer, H., Simplicial function spaces and simplexes, Expo. math., 3, 165-168, (1985), MR 87c:46009 · Zbl 0564.46007 [6] Bernstein, S., Sur LES fonctions absolument monotones, Acta math., 51, 1-66, (1928) · JFM 55.0142.07 [7] Bliedtner, J.; Hansen, W., Simplicial cones in potential theory, Inventiones math., 29, 83-110, (1975), MR 52 #8470 · Zbl 0308.31011 [8] Bliedtner, J.; Hansen, W., The weak Dirichlet problem, J. reine angew. math., 348, 34-39, (1984), MR 85h:31012 · Zbl 0536.31009 [9] Bliedtner, J.; Hansen, W., Potential theory — an analytic and probabilistic approach to balayage, (1986), Springer-Verlag London, MR 88b:31002 · Zbl 0706.31001 [10] Bochner, S., Harmonic analysis and the theory of probability, (1955), University of California Press Berlin, MR 17 #273d · Zbl 0068.11702 [11] Caffarelli, L.A.; Littman, W., Representation formulas for solutions to δu − u = 0 in ℝ^{n}, (), 249-263, MR 84k:35045 [12] Choquet, G., Lectures on analysis I - III, (1969), W. A. Benjamin, Inc. Washington, D.C., MR 40 #3254 [13] Choquet, G., Deux exemples classiques de représentation integrale, Enseignement math., 15, 2, 63-75, (1969), MR 40 #6224 · Zbl 0175.42202 [14] Edgar, G.A., Two integral representations, (), 193-198, MR 85g:30034 [15] Effros, E.G.; Kazdan, J.L., Applications of Choquet simplexes to elliptic and parabolic boundary value problems, J. diff. eq., 8, 95-134, (1970), MR 41 #4215 · Zbl 0255.46018 [16] Fonf, V.P.; Lindenstrauss, J.; Phelps, R.R., Infinite dimensional convexity, () · Zbl 1086.46004 [17] Hansen, W., A Liouville property for spherical averages in the plane, () · Zbl 1006.31001 [18] Hansen, W.; Nadirashvili, N., Littlewood’s one circle problem, J. London math. soc., 50, 2, 349-360, (1994), MR 95j:31002 · Zbl 0804.31001 [19] Hansen, W.; Nadirashvili, N., On Veech’s conjecture for harmonic functions, Ann. scuola norm. sup. Pisa cl.-sci., 22, 4, 137-153, (1995), MR 96c:31004 · Zbl 0846.31003 [20] Helms, L.L., Introduction to potential theory, (), MR 41 #5638 [21] Holland, F., The extreme points of a class of functions with positive real part, Math. ann., 202, 85-87, (1973), MR 49 #562 · Zbl 0246.30027 [22] Hunt, R.R.; Wheeden, R.L., Positive harmonic functions on Lipschitz domains, Trans. amer. math. soc., 147, 505-527, (1970), MR 43 #547 · Zbl 0193.39601 [23] Jacobs, K., Extremalpunkte konvexer mengen, Selecta math., heidelberger taschenbucher, 86, 90-118, (1971), MR 58 #30754 · Zbl 0219.46014 [24] Keldysh, M.V., On the solubility and stability of the Dirichlet problem (Russian), Uspechi mat. nauk., 8, 171-292, (1941), MR 3 #123f [25] Keldysh, M.V., On the Dirichlet problem (Russian), Dokl. akad. nauk SSSR, 32, 308-309, (1941), MR 6 #64a · Zbl 0061.23104 [26] Klee, V., Some new results on smoothness and rotundity in normed linear spaces, Math. ann., 139, 51-63, (1959), MR 22 #5879 · Zbl 0092.11602 [27] Korányi, A., A survey of harmonic functions on symmetric spaces, (), 323-344, MR 80k:43012 [28] Král, J.; Netuka, I.; Veselý, J., Potential theory IV (Czech), (1977), SPN New York-London-Sydney [29] Kružík, M., Bauer’s maximum principle and hulls of sets, Calc. var. partial differential equations, 11, 321-332, (2000), MR 2001k:49005 · Zbl 0981.49010 [30] Lindenstrauss, J., Some useful facts about Banach spaces, (), 185-200, MR 89g:46015 · Zbl 0651.46019 [31] Lukeš, J.; Malý, J., Measure and integral, (1995), Matfyzpress Berlin · Zbl 0888.28001 [32] Lukeš, J.; Malý, J.; Zajíček, L., Fine topology methods in real analysis and potential theory, (), MR 89b:31001 · Zbl 0607.31001 [33] Martin, R.S., Minimal positive harmonic functions, Trans. amer. math. soc., 49, 137-172, (1941), MR 2 #292h · JFM 67.0343.03 [34] Netuka, I., The Dirichlet problem for harmonic functions, Amer. math. monthly, 87, 621-628, (1980), MR 82c:31005 · Zbl 0454.31002 [35] Netuka, I.; Veselý, J., Dirichlet problem and the Keldysh theorem (Czech), Pokroky mat. fyz. astronom., 24, 77-88, (1979), MR 82f:01126 [36] Netuka, I.; Veselý, J., Mean value property and harmonic functions, (), 359-398, MR 96c:31001 · Zbl 0863.31012 [37] Phelps, R.R., Lectures on Choquet’s theorem, (1966), D. Van Nostrand Co., Inc. Dordrecht, MR 33 #1690, (2nd ed.: Springer Verlag, Berlin, 2001) · Zbl 0135.36203 [38] Price, G.B., On the extreme points of convex sets, Duke math. J., 3, 56-67, (1937) · JFM 63.0668.05 [39] Rakestraw, R.M., A representation theorem for real convex functions, Pac. J. math., 60, 165-168, (1975), MR 52 #14193 · Zbl 0266.26009 [40] Robertson, M.S., On the coefficients of a typically-real function, Bul. amer. math. soc., 41, 565-572, (1935) · JFM 61.0348.03 [41] Roubíček, T., Relaxation in optimization theory and variational calculus, (), MR 98e:49002 · Zbl 0880.49002 [42] Veech, W.A., A converse to the Mean value theorem for harmonic functions, Amer. J. math., 97, 1007-1027, (1975), MR 52 #14330 · Zbl 0324.31002
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