The obstacle problem for the fractional Laplacian with critical drift

Abstract

We study the obstacle problem for the fractional Laplacian with drift, $\min \left\{(-\Delta {)}^{s}u+b\cdot \nabla u,u-\phi \right\}=0$ in ${\mathbb{R}}^n$, in the critical regime $s=\frac{1}{2}$. Our main result establishes the $C^{1,\alpha }$ regularity of the free boundary around any regular point $x_0$, with an expansion of the form \[ u(x)-φ(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tildeγ(x_0)} + o\left(|x-x_0|^{1+\tildeγ(x_0)+σ}\right), \] \[ \tildeγ(x_0) = \frac{1}{2}+\frac{1}π \arctan (b\cdot e), \] where $e\in {\mathbb{S}}^{n-1}$ is the normal vector to the free boundary, $\sigma >0$, and $c_0>0$. We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent $\tilde{\gamma }(x_0)$ also depends on the operator.