Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 33, 2008, 523-548

# REGULARITY AND FREE BOUNDARY
REGULARITY FOR THE *p* LAPLACIAN
IN LIPSCHITZ AND *C*^{1} DOMAINS

## John L. Lewis and Kaj Nyström

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 Umeå, 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 *C*^{1} domains. To
formulate our results we let \Omega \subset **R**^{n}
be a bounded Lipschitz domain with
constant *M*. Given *p*, 1 p w \in \partial\Omega, 0 r r_{0},
suppose that *u* is a positive *p* harmonic function
in \Omega \cap *B*(*w*,4*r*), that *u* is
continuous in \bar\Omega \cap \bar*B*(*w*,4*r*)
and *u* = 0 on \Delta(*w*,4*r*). We
first prove, Theorem 1, that \nabla*u*(*y*) \to
\nabla*u*(*x*), for almost every *x* \in
\Delta(*w*,4*r*), as *y* \to *x* non
tangentially in \Omega. Moreover,
||log|\nabla*u*|||_{BMO(\Delta(w,r))}
\leq *c*(*p*,*n*,*M*). If, in addition,
\Omega is *C*^{1} regular then we prove, Theorem 2,
that log|\nabla*u*| \in *VMO*(\Delta(*w*,*r*)).
Finally we prove, Theorem 3, that there exists \hat*M*,
independent of *u*, such that if *M* \leq \hat*M* and if
log|\nabla*u*| \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. Nyström:
Regularity and free boundary regularity for
the *p* Laplacian in Lipschitz and *C*^{1} domains.
Ann. Acad. Sci. Fenn. Math. 33 (2008), 523-548.

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