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

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- by John L. Lewis and Kaj Nyström;
- J. Amer. Math. Soc.
**25**(2012), 827-862 - DOI: https://doi.org/10.1090/S0894-0347-2011-00726-1
- Published electronically: December 8, 2011
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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, $\log | \nabla u |$ is of bounded mean oscillation on $\partial \Omega \cap B (w, r)$ with $\| \log | \nabla u |\|_{\mathrm {BMO} (\partial \Omega \cap B(w, r))} \leq c$. If, in addition, $\Omega$ is Reifenberg flat with vanishing constant and $n\in \mathrm {VMO}(\partial \Omega \cap B(w, 4r))$, where $n$ denotes the unit inner normal to $\partial \Omega$ in the measure-theoretic sense, then in Theorem 2 we prove that $\log | \nabla u | \in \mathrm {VMO}(\partial \Omega \cap B(w, r))$. In Theorem 3 we prove the following converse to Theorem 2. Suppose $u$ is as in Theorem 1, $\log | \nabla u | \in \mathrm {VMO}(\partial \Omega \cap B(w, r))$, 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 $n\in \mathrm {VMO}(\partial \Omega \cap B(w, r/2))$. Finally, in Theorem 4 we establish a two-phase version of Theorem 3 without the smallness assumption on $\delta .$## References

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## Bibliographic Information

**John L. Lewis**- Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506-0027
- Email: john@ms.uky.edu
**Kaj Nyström**- Affiliation: Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden
- Email: kaj.nystrom@math.uu.se
- 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 - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**25**(2012), 827-862 - MSC (2010): Primary 35J25, 35J70
- DOI: https://doi.org/10.1090/S0894-0347-2011-00726-1
- MathSciNet review: 2904575