Almost everywhere regularity for the free boundary of the 𝑝-harmonic obstacle problem 𝑝>2

Let u u be a solution to the normalized p p -harmonic obstacle problem with p > 2 p>2 . That is, u ∈ W 1 , p ( B 1 ( 0 ) ) u\in W^{1,p}(B_1(0)) , 2 > p > ∞ 2>p>\infty , u ≥ 0 u\ge 0 and d i v ( | ∇ u | p − 2 ∇ u ) = χ { u > 0 }    in    B 1 ( 0 ) \begin{equation*} \mathrm {div}( |\nabla u|^{p-2}\nabla u)=\chi _{\{u>0\}} \ \text { in } \ B_1(0) \end{equation*} where u ( x ) ≥ 0 u(x)\ge 0 and χ A \chi _A is the characteristic function of the set A A . The main result is that for almost every free boundary point with respect to the ( n − 1 ) (n-1) -Hausdorff measure, there is a neighborhood where the free boundary is a C 1 , β C^{1,\beta } -graph. That is, for H n − 1 \mathcal {H}^{n-1} -a.e. point x 0 ∈ ∂ { u > 0 } ∩ B 1 ( 0 ) x^0\in \partial \{u>0\}\cap B_1(0) there is an r > 0 r>0 such that B r ( x 0 ) ∩ ∂ { u > 0 } ∈ C 1 , β B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta } .