Boundary Behaviour for p Harmonic Functions in Lipschitz and Starlike Lipschitz Ring Domains

In this paper we prove new results for p harmonic functions, p 6 = 2, 1 < p < ∞, 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 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, Theorem 3 and Theorem 4, appropriate extensions to p 6 = 2, 1 < p <∞, of famous results of Dahlberg [D] and Jerison-Kenig [JK1] on the Poisson kernel associated to the Laplace operator (i.e. p = 2)