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

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 $\delta .$