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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:
  Alfsen, E.M., Compact convex sets and boundary integrals, (1971), Springer-Verlag, MR 56 #3615 · Zbl 0209.42601  Armitage, D.H., The Riesz-Herglotz representation for positive harmonic functions via Choquet’s theorem, (), 229-232, MR 97f:31006 · Zbl 0856.31003  Armitage, D.H.; Gardiner, S.J., Classical potential theory, (2001), Springer-Verlag London, Ltd. Berlin, MR 2001m:31001 · Zbl 0972.31001  Bauer, H., Approximation and abstract boundaries, Amer. math. monthly, 85, 632-647, (1978), MR 80f:41014 · Zbl 0416.41023  Bauer, H., Simplicial function spaces and simplexes, Expo. math., 3, 165-168, (1985), MR 87c:46009 · Zbl 0564.46007  Bernstein, S., Sur LES fonctions absolument monotones, Acta math., 51, 1-66, (1928) · JFM 55.0142.07  Bliedtner, J.; Hansen, W., Simplicial cones in potential theory, Inventiones math., 29, 83-110, (1975), MR 52 #8470 · Zbl 0308.31011  Bliedtner, J.; Hansen, W., The weak Dirichlet problem, J. reine angew. math., 348, 34-39, (1984), MR 85h:31012 · Zbl 0536.31009  Bliedtner, J.; Hansen, W., Potential theory — an analytic and probabilistic approach to balayage, (1986), Springer-Verlag London, MR 88b:31002 · Zbl 0706.31001  Bochner, S., Harmonic analysis and the theory of probability, (1955), University of California Press Berlin, MR 17 #273d · Zbl 0068.11702  Caffarelli, L.A.; Littman, W., Representation formulas for solutions to δu − u = 0 in ℝ^{n}, (), 249-263, MR 84k:35045  Choquet, G., Lectures on analysis I - III, (1969), W. A. Benjamin, Inc. Washington, D.C., MR 40 #3254  Choquet, G., Deux exemples classiques de représentation integrale, Enseignement math., 15, 2, 63-75, (1969), MR 40 #6224 · Zbl 0175.42202  Edgar, G.A., Two integral representations, (), 193-198, MR 85g:30034  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  Fonf, V.P.; Lindenstrauss, J.; Phelps, R.R., Infinite dimensional convexity, () · Zbl 1086.46004  Hansen, W., A Liouville property for spherical averages in the plane, () · Zbl 1006.31001  Hansen, W.; Nadirashvili, N., Littlewood’s one circle problem, J. London math. soc., 50, 2, 349-360, (1994), MR 95j:31002 · Zbl 0804.31001  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  Helms, L.L., Introduction to potential theory, (), MR 41 #5638  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  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  Jacobs, K., Extremalpunkte konvexer mengen, Selecta math., heidelberger taschenbucher, 86, 90-118, (1971), MR 58 #30754 · Zbl 0219.46014  Keldysh, M.V., On the solubility and stability of the Dirichlet problem (Russian), Uspechi mat. nauk., 8, 171-292, (1941), MR 3 #123f  Keldysh, M.V., On the Dirichlet problem (Russian), Dokl. akad. nauk SSSR, 32, 308-309, (1941), MR 6 #64a · Zbl 0061.23104  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  Korányi, A., A survey of harmonic functions on symmetric spaces, (), 323-344, MR 80k:43012  Král, J.; Netuka, I.; Veselý, J., Potential theory IV (Czech), (1977), SPN New York-London-Sydney  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  Lindenstrauss, J., Some useful facts about Banach spaces, (), 185-200, MR 89g:46015 · Zbl 0651.46019  Lukeš, J.; Malý, J., Measure and integral, (1995), Matfyzpress Berlin · Zbl 0888.28001  Lukeš, J.; Malý, J.; Zajíček, L., Fine topology methods in real analysis and potential theory, (), MR 89b:31001 · Zbl 0607.31001  Martin, R.S., Minimal positive harmonic functions, Trans. amer. math. soc., 49, 137-172, (1941), MR 2 #292h · JFM 67.0343.03  Netuka, I., The Dirichlet problem for harmonic functions, Amer. math. monthly, 87, 621-628, (1980), MR 82c:31005 · Zbl 0454.31002  Netuka, I.; Veselý, J., Dirichlet problem and the Keldysh theorem (Czech), Pokroky mat. fyz. astronom., 24, 77-88, (1979), MR 82f:01126  Netuka, I.; Veselý, J., Mean value property and harmonic functions, (), 359-398, MR 96c:31001 · Zbl 0863.31012  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  Price, G.B., On the extreme points of convex sets, Duke math. J., 3, 56-67, (1937) · JFM 63.0668.05  Rakestraw, R.M., A representation theorem for real convex functions, Pac. J. math., 60, 165-168, (1975), MR 52 #14193 · Zbl 0266.26009  Robertson, M.S., On the coefficients of a typically-real function, Bul. amer. math. soc., 41, 565-572, (1935) · JFM 61.0348.03  Roubíček, T., Relaxation in optimization theory and variational calculus, (), MR 98e:49002 · Zbl 0880.49002  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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