Abstract:Let $u$ be a solution to the normalized $p$-harmonic obstacle problem with $p>2$. That is, $u\in W^{1,p}(B_1(0))$, $2<p<\infty$, $u\ge 0$ and \begin{equation*} \mathrm {div}( |\nabla u|^{p-2}\nabla u)=\chi _{\{u>0\}} \ \text { in } \ B_1(0) \end{equation*} 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 $B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta }$.

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中文翻译：

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harm-谐波障碍问题𝑝> 2的自由边界的几乎所有地方的规则性

摘要：让$ u $是具有$ p> 2 $的归一化$ p $-谐波障碍问题的解决方案。也就是$ u \ in W ^ {1，p}（B_1（0））$，$ 2 <p <\ infty $，$ u \ ge 0 $和\ begin {equation *} \ mathrm {div}（| \ nabla u | ^ {p-2} \ nabla u）= \ chi _ {\ {u> 0 \}} \ \ text {in} \ B_1（0）\ end {equation *}其中$ u（x） \ ge 0 $和$ \ chi _A $是集合$ A $的特征函数。主要结果是，对于$（n-1）$-Hausdorff度量，几乎每个自由边界点都有一个邻域，其中自由边界是$ C ^ {1，\ beta} $图。也就是说，对于$ \ mathcal {H} ^ {n-1} $-ae点$ x ^ 0 \在\ partial \ {u> 0 \} \ cap B_1（0）$中，有一个$ r> 0 $这样C ^ {1，\ beta} $中的$ B_r（x ^ 0）\ cap \ partial \ {u> 0 \} \。

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