We study the obstacle problem for integro-differential operators of order$2s$, with $s\in (0,1)$. Our main result establishes that the free boundary is$C^{1,\gamma}$ and $u\in C^{1,s}$ near all regular points. Namely, we prove thefollowing dichotomy at all free boundary points $x_0\in\partial\{u=\varphi\}$: (i) either $u(x)-\varphi(x)=c\,d^{1+s}(x)+o(|x-x_0|^{1+s+\alpha})$ for some$c>0$, (ii) or $u(x)-\varphi(x)=o(|x-x_0|^{1+s+\alpha})$, where $d$ is the distance to the contact set $\{u=\varphi\}$. Moreover, weshow that the set of free boundary points $x_0$ satisfying (i) is open, andthat the free boundary is $C^{1,\gamma}$ and $u\in C^{1,s}$ near those points. These results were only known for the fractional Laplacian \cite{CSS}, andare completely new for more general integro-differential operators. The methodswe develop here are purely nonlocal, and do not rely on any monotonicity-typeformula for the operator. Thanks to this, our techniques can be applied in themuch more general context of fully nonlinear integro-differential operators: weestablish similar regularity results for obstacle problems with convexoperators.
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机译：我们研究了阶为$ 2s $且$ s \ in（0,1）$的积分微分算子的障碍问题。我们的主要结果表明，自由边界在所有正则点附近为$ C ^ {1，\ gamma} $和$ u \ in C ^ {1，s} $。即，我们证明了在所有自由边界点$ x_0 \ in \ partial \ {u = \ varphi \} $之后的二分法：（i）$ u（x）-\ varphi（x）= c \，d ^ {1 + s}（x）+ o（| x-x_0 | ^ {1 + s + \ alpha}）$ for $$ c> 0 $，（ii）或$ u（x）-\ varphi（x）= o（ | x-x_0 | ^ {1 + s + \ alpha}）$，其中$ d $是到联系人集$ \ {u = \ varphi \} $的距离。此外，我们证明满足（i）的自由边界点$ x_0 $的集合是开放的，并且自由边界点附近的是$ C ^ {1，\ gamma} $和$ u \ in C ^ {1，s} $点。这些结果仅对于分数Laplacian \ cite {CSS}为人所知，对于更一般的积分微分算子来说是全新的。我们在这里开发的方法纯粹是非局部的，对于操作者而言，它不依赖于任何单调性式。由于这个原因，我们的技术可以在完全非线性积分微分算子的更一般的上下文中应用：对于凸算子的障碍问题，我们建立了相似的规律性结果。
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