Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Regularity and free boundary regularity for the $ p$-Laplace operator in Reifenberg flat and Ahlfors regular domains

Authors: John L. Lewis and Kaj Nyström
Journal: J. Amer. Math. Soc. 25 (2012), 827-862
MSC (2010): Primary 35J25, 35J70
Published electronically: December 8, 2011
MathSciNet review: 2904575
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Abstract: In this paper we solve several problems concerning regularity and free boundary regularity, below the continuous threshold, for positive solutions to the $ p$-Laplace equation, $ 1 < p < \infty $, vanishing on a portion of the boundary of an Ahlfors regular NTA-domain. In Theorem 1 of our paper we show that if $ \Omega \subset \mathbf {R}^{n}, n \geq 2, $ is an Ahlfors regular NTA-domain and $ u$ is a positive $ p$-harmonic function in $ \Omega \cap B (w, 4r) $, with continuous boundary value 0 on $ \partial \Omega \cap B (w, 4r)$, then $ \nabla u (x) \to \nabla u (y) $ nontangentially as $ x \rightarrow y \in \partial \Omega \cap B (w, 4r),$ almost everywhere with respect to surface area, $ \sigma , $ on $ \partial \Omega \cap B (w, 4 r). $ Moreover, $ \log \vert \nabla u \vert $ is of bounded mean oscillation on $ \partial \Omega \cap B (w, r) $ with $ \Vert \log \vert \nabla u \vert\Vert _{\textup {BMO} (\partial \Omega \cap B(w, r))} \leq c$. If, in addition, $ \Omega $ is Reifenberg flat with vanishing constant and $ n\in \textup {VMO}(\partial \Omega \cap B(w, 4r))$, where $ n $ denotes the unit inner normal to $ \partial \Omega $ in the measure-theoretic sense, then in Theorem 2 we prove that $ \log \vert \nabla u \vert \in \textup {VMO}(\partial \Omega \cap B(w, r))$. In Theorem 3 we prove the following converse to Theorem 2. Suppose $ u$ is as in Theorem 1, $ \log \vert \nabla u \vert \in \textup {VMO}(\partial \Omega \cap B(w, r))$, and that $ \partial \Omega \cap B (w, r) $ is $ (\delta , r_0)$-Reifenberg flat. Then there exists $ \bar \delta = \bar \delta (p, n) $ such that if $ 0 < \delta \leq \bar \delta , $ then $ \partial \Omega \cap B(w, r/2) $ is Reifenberg flat with vanishing constant and $ n\in \textup {VMO}(\partial \Omega \cap B(w, r/2))$. Finally, in Theorem 4 we establish a two-phase version of Theorem 3 without the smallness assumption on $ \delta .$

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John L. Lewis
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027

Kaj Nyström
Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden

Keywords: $p$-harmonic function, Reifenberg flat domain, Ahlfors regular domain, regularity, free boundary regularity
Received by editor(s): June 13, 2011
Received by editor(s) in revised form: July 21, 2011
Published electronically: December 8, 2011
Additional Notes: The first author was partially supported by NSF DMS-0900291
The second author was partially supported by grant VR-70629701 from the Swedish research council VR
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.