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 1 < p < \infty \Omega \subset \mathbf {R}^{n}, n \geq 2, is a positive p \Omega \cap B (w, 4r) \partial \Omega \cap B (w, 4r) \nabla u (x) \to \nabla u (y) x \rightarrow y \in \partial \Omega \cap B (w, 4r), on \partial \Omega \cap B (w, 4 r). \log \vert \nabla u \vert \partial \Omega \cap B (w, r) \Vert \log \vert \nabla u \vert\Vert _{\textup {BMO} (\partial \Omega \cap B(w, r))} \leq c is Reifenberg flat with vanishing constant and n\in \textup {VMO}(\partial \Omega \cap B(w, 4r)) denotes the unit inner normal to \partial \Omega \log \vert \nabla u \vert \in \textup {VMO}(\partial \Omega \cap B(w, r)) is as in Theorem 1, \log \vert \nabla u \vert \in \textup {VMO}(\partial \Omega \cap B(w, r)) \partial \Omega \cap B (w, r) (\delta , r_0) \bar \delta = \bar \delta (p, n) 0 < \delta \leq \bar \delta , \partial \Omega \cap B(w, r/2) n\in \textup {VMO}(\partial \Omega \cap B(w, r/2))