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Zbl 1134.31008
Lewis, John L.; Nyström, Kaj
Boundary behaviour for $p$ harmonic functions in Lipschitz and starlike Lipschitz ring domains. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 40, No. 5, 765-813 (2007). ISSN 0012-9593

Summary: In this paper we prove new results for $p$ harmonic functions, $p\neq 2$, $1<p<\infty $, in Lipschitz and starlike Lipschitz ring domains. In particular we prove the boundary Harnack inequality, Theorem 1, for the ratio of two positive $p$ harmonic functions vanishing on a portion of the boundary of a Lipschitz domain, with constants only depending on $p,n$ and the Lipschitz constant of the domain. For $p$ capacitary functions, in starlike Lipschitz ring domains, we prove an even stronger result, Theorem 2, showing that the ratio is Hölder continuous up to the boundary. Moreover, for $p$ capacitary functions in starlike Lipschitz ring domains we prove, Theorems 3 and 4, appropriate extensions to $p\neq 2$, $1<p<\infty $, of famous results of {\it B. Dahlberg} [Arch. Ration. Mech. Anal. 65, 275--288 (1977; Zbl 0406.28009)] and {\it D. S. Jerison} and {\it C. E. Kenig} [Trans. Am. Math. Soc. 273, 781--794 (1982; Zbl 0494.31003)] on the Poisson kernel associated to the Laplace operator (i.e. $p=2$).

MSC 2000:: *31B25 Boundary behavior of harmonic functions (higher-dim.)
58E20 Harmonic maps between infinite-dimensional spaces

Citations: Zbl 0406.28009; Zbl 0494.31003

Cited in: Zbl 1241.35110 Zbl 1210.31004

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