Uniform boundary Harnack principle and generalized triangle property

Hansen W (2005)
Journal of Functional Analysis 226(2): 452-484.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Abstract / Bemerkung
Let D be a bounded open subset in R-d, d >= 2 and let G denote the Green function for D with respect to (-Delta)(alpha/2), 0 < alpha <= 2, alpha < d. If alpha < 2, assume that D satisfies the interior corkscrew condition-, if alpha=2, i.e., if G is the classical Green function on D, assume-more restrictively-that D is a uniform domain. Let g = G((.), y(0)) Lambda 1 for some y(0) is an element of D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that G(z, y)/g(z) <= C G(x, y)/g(x) when d(z, x) <= d(z, y). An intermediate step is the approximation G(x, y) approximate to vertical bar x - y vertical bar(alpha-d)g(x)g(y)/g(A)(2), where A is an arbitrary point in a certain set B(x, y). This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D x D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent. (c) 2005 Elsevier Inc. All rights reserved.
Stichworte
quasi-metric property; approximation of Green function; comparison of Green functions; laplace operator; fractional Laplacian; riesz potentials; hamack chain; Corkscrew condition; uniform domain; NTA-domam; property; generalized triangle; boundary Harnack principle; green function
Erscheinungsjahr
2005
Zeitschriftentitel
Journal of Functional Analysis
Band
226
Ausgabe
2
Seite(n)
452-484
ISSN
0022-1236
Page URI
https://pub.uni-bielefeld.de/record/1602338

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Hansen W. Uniform boundary Harnack principle and generalized triangle property. Journal of Functional Analysis. 2005;226(2):452-484.
Hansen, W. (2005). Uniform boundary Harnack principle and generalized triangle property. Journal of Functional Analysis, 226(2), 452-484. doi:10.1016/j.jfa.2004.12.010
Hansen, W. (2005). Uniform boundary Harnack principle and generalized triangle property. Journal of Functional Analysis 226, 452-484.
Hansen, W., 2005. Uniform boundary Harnack principle and generalized triangle property. Journal of Functional Analysis, 226(2), p 452-484.
W. Hansen, “Uniform boundary Harnack principle and generalized triangle property”, Journal of Functional Analysis, vol. 226, 2005, pp. 452-484.
Hansen, W.: Uniform boundary Harnack principle and generalized triangle property. Journal of Functional Analysis. 226, 452-484 (2005).
Hansen, Wolfhard. “Uniform boundary Harnack principle and generalized triangle property”. Journal of Functional Analysis 226.2 (2005): 452-484.