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

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 $\delta .$