| Title:
|
On the Dirichlet problem for functions of the first Baire class (English) |
| Author:
|
Spurný, Jiří |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
721-728 |
| . |
| Category:
|
math |
| . |
| 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 $X$. (English) |
| Keyword:
|
weak Dirichlet problem |
| Keyword:
|
function space |
| Keyword:
|
Choquet simplexes |
| Keyword:
|
Baire-one functions |
| MSC:
|
26A21 |
| MSC:
|
31B05 |
| MSC:
|
31C45 |
| MSC:
|
46A55 |
| idZBL:
|
Zbl 1090.46500 |
| idMR:
|
MR1883380 |
| . |
| Date available:
|
2009-01-08T19:18:06Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119287 |
| . |
| Reference:
|
[1] Alfsen E.M.: Compact convex sets and boundary integrals.Springer-Verlag New York-Heidelberg (1971). Zbl 0209.42601, MR 0445271 |
| Reference:
|
[2] Bauer H.: Axiomatische behandlung des Dirichletschen problems fur elliptische und parabolische differentialgleichungen.Math. Ann. 146 (1962), 1-59. MR 0144064 |
| Reference:
|
[3] Boboc N., Cornea A.: Convex cones of lower semicontinuous functions on compact spaces.Rev. Roum. Math. Pures. App. 12 (1967), 471-525. Zbl 0155.17301, MR 0216278 |
| Reference:
|
[4] Bliedtner J., Hansen W.: Simplicial cones in potential theory.Invent. Math. (2) 29 (1975), 83-110. Zbl 0308.31011, MR 0387630 |
| Reference:
|
[5] Capon M.: Sur les fonctions qui vérifient le calcul barycentrique.Proc. London Math. Soc. (3) 32 (1976), 163-180. Zbl 0313.46003, MR 0394148 |
| Reference:
|
[6] Engelking R.: General Topology.Heldermann, Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[7] Choquet G.: Lectures on analysis vol. II: Representation theory.W.A. Benjamin, Inc., New York-Amsterdam (1969). Zbl 0181.39602, MR 0250012 |
| Reference:
|
[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. Zbl 0582.46010, MR 0780351 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[11] Phelps R.R.: Lectures on Choquet's theorem.D. Van Nostrand Co. (1966). Zbl 0135.36203, MR 0193470 |
| . |
