Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions
weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions
Summary:
Let $\Cal H$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname{Ch}X$ of $\Cal H$ is an $F_\sigma$-set if and only if given any bounded Baire-one function $f$ on $\operatorname{Ch}X$ there is an $\Cal H$-affine bounded Baire-one function $h$ on $X$ such that $h=f$ on $\operatorname{Ch}X$. This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$.
References:
[1] Alfsen E.M.: Compact convex sets and boundary integrals. Springer-Verlag New York-Heidelberg (1971). MR 0445271 | Zbl 0209.42601
[2] Bauer H.: Axiomatische behandlung des Dirichletschen problems fur elliptische und parabolische differentialgleichungen. Math. Ann. 146 (1962), 1-59. MR 0144064
[3] Boboc N., Cornea A.: Convex cones of lower semicontinuous functions on compact spaces. Rev. Roum. Math. Pures. App. 12 (1967), 471-525. MR 0216278 | Zbl 0155.17301
[4] Bliedtner J., Hansen W.: Simplicial cones in potential theory. Invent. Math. (2) 29 (1975), 83-110. MR 0387630 | Zbl 0308.31011
[5] Capon M.: Sur les fonctions qui vérifient le calcul barycentrique. Proc. London Math. Soc. (3) 32 (1976), 163-180. MR 0394148 | Zbl 0313.46003
[6] Engelking R.: General Topology. Heldermann, Berlin (1989). MR 1039321 | Zbl 0684.54001
[7] Choquet G.: Lectures on analysis vol. II: Representation theory. W.A. Benjamin, Inc., New York-Amsterdam (1969). MR 0250012 | Zbl 0181.39602
[8] Jellett F.: On affine extensions of continuous functions defined on the extreme boundary of a Choquet simplex. Quart. J. Math. Oxford (2) 36 (1985), 71-73. MR 0780351 | Zbl 0582.46010
[9] Lacey H.E.. Morris P.D.: On spaces of type $A(K)$ and their duals. Proc. Amer. Math. Soc. 23 (1969), 151-157. MR 0625855
[10] Lukeš J., Malý J., Zajíček L.: Fine topology methods in real analysis and potential theory. Lecture Notes in Math. 1189 Springer-Verlag (1986). MR 0861411
[11] Phelps R.R.: Lectures on Choquet's theorem. D. Van Nostrand Co. (1966). MR 0193470 | Zbl 0135.36203
Search
Partner of
