Annales AcademiŠ Scientiarum FennicŠ
Mathematica
Volumen 33, 2008, 523-548

REGULARITY AND FREE BOUNDARY REGULARITY FOR THE p LAPLACIAN IN LIPSCHITZ AND C1 DOMAINS

John L. Lewis and Kaj Nystr÷m

University of Kentucky, Department of Mathematics
Lexington, KY 40506-0027, U.S.A.; john 'at' ms.uky.edu

Ume{\aa} University, Department of Mathematics
S-90187 Umeň, Sweden; kaj.nystrom 'at' math.umu.se

Abstract. In this paper we study regularity and free boundary regularity, below the continuous threshold, for the p Laplace equation in Lipschitz and C1 domains. To formulate our results we let \Omega \subset Rn be a bounded Lipschitz domain with constant M. Given p, 1 p w \in \partial\Omega, 0 r r0, suppose that u is a positive p harmonic function in \Omega \cap B(w,4r), that u is continuous in \bar\Omega \cap \barB(w,4r) and u = 0 on \Delta(w,4r). We first prove, Theorem 1, that \nablau(y) \to \nablau(x), for almost every x \in \Delta(w,4r), as y \to x non tangentially in \Omega. Moreover, ||log|\nablau|||BMO(\Delta(w,r)) \leq c(p,n,M). If, in addition, \Omega is C1 regular then we prove, Theorem 2, that log|\nablau| \in VMO(\Delta(w,r)). Finally we prove, Theorem 3, that there exists \hatM, independent of u, such that if M \leq \hatM and if log|\nablau| \in VMO(\Delta(w,r)) then the outer unit normal to \partial\Omega, n, is in VMO(\Delta(w,r/2)).

2000 Mathematics Subject Classification: Primary 35J25, 35J70.

Key words: p harmonic function, Lipschitz domain, regularity, free boundary regularity, elliptic measure, blow-up.

Reference to this article: J.L. Lewis and K. Nystr÷m: Regularity and free boundary regularity for the p Laplacian in Lipschitz and C1 domains. Ann. Acad. Sci. Fenn. Math. 33 (2008), 523-548.

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