On Evans' and Choquet's Theorems for Polar Sets

Hansen W, Netuka I (2021)
Potential Analysis 2022(56): 423–43.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Hansen, WolfhardUniBi; Netuka, Ivan
Abstract / Bemerkung
By classical results of G.C. Evans and G. Choquet on "good" kernels G in potential theory, for every polar K-sigma-set P, there exists a finite measure mu on P such that its potential G mu is infinite on P, and a set P admits a finite measure mu on P such that G mu is infinite exactly on P if and only if P is a polar G(delta)-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. It is shown that, by an elementary "metric sweeping" of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = vertical bar x-y vertical bar(alpha-d) on R-d, 2 < alpha < d, solves a long-standing open problem.
Stichworte
Polar set; Capacity; Potential
Erscheinungsjahr
2021
Zeitschriftentitel
Potential Analysis
Band
2022
Ausgabe
56
Seite(n)
423–43
ISSN
0926-2601
eISSN
1572-929X
Finanzierungs-Informationen
Open-Access-Publikationskosten wurden durch die Universität Bielefeld im Rahmen des DEAL-Vertrags gefördert.
Page URI
https://pub.uni-bielefeld.de/record/2953399

Zitieren

Hansen W, Netuka I. On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis. 2021;2022(56):423–43.
Hansen, W., & Netuka, I. (2021). On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis, 2022(56), 423–43. https://doi.org/10.1007/s11118-020-09890-0
Hansen, Wolfhard, and Netuka, Ivan. 2021. “On Evans' and Choquet's Theorems for Polar Sets”. Potential Analysis 2022 (56): 423–43.
Hansen, W., and Netuka, I. (2021). On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis 2022, 423–43.
Hansen, W., & Netuka, I., 2021. On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis, 2022(56), p 423–43.
W. Hansen and I. Netuka, “On Evans' and Choquet's Theorems for Polar Sets”, Potential Analysis, vol. 2022, 2021, pp. 423–43.
Hansen, W., Netuka, I.: On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis. 2022, 423–43 (2021).
Hansen, Wolfhard, and Netuka, Ivan. “On Evans' and Choquet's Theorems for Polar Sets”. Potential Analysis 2022.56 (2021): 423–43.
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2022-08-15T09:09:47Z
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