Geodesic
Almost everywhere regularity for the free boundary of the $p$-harmonic obstacle problem $p>2$
Algebra i analiz, Tome 32 (2020) no. 3, pp. 39-64.

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Let $u$ be a solution to the normalized $p$-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2$, $u\ge 0$ and $$ \mathrm{div}\,( |\nabla u|^{p-2}\nabla u)=\chi_{\{u>0\}}\textrm{ in }B_1(0) $$ where $u(x)\ge 0$ and $\chi_A$ is the characteristic function of the set $A$. The main result is that for almost every free boundary point with respect to the $(n-1)$-Hausdorff measure, there is a neighborhood where the free boundary is a $C^{1,\beta}$-graph. That is, for $\mathcal{H}^{n-1}$-a.e. point $x^0\in \partial \{u>0\}\cap B_1(0)$ there is an $r>0$ such that $B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta}$.
Keywords: $p$-Laplace operator, blow-up, Carleson measure Hausdorff measure. 
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J. Andersson. Almost everywhere regularity for the free boundary of the $p$-harmonic obstacle problem $p>2$. Algebra i analiz, Tome 32 (2020) no. 3, pp. 39-64. http://geodesic.mathdoc.fr/item/AA_2020_32_3_a2/

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