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Boundary behaviour for \(p\) harmonic functions in Lipschitz and starlike Lipschitz ring domains. (English) Zbl 1134.31008

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 B. Dahlberg [Arch. Ration. Mech. Anal. 65, 275–288 (1977; Zbl 0406.28009)] and D. S. Jerison and 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:

31B25 Boundary behavior of harmonic functions in higher dimensions
58E20 Harmonic maps, etc.
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