The obstacle problem for the fractional Laplacian with critical drift

TL;DR

This paper proves the $C^{1,eta}$ regularity of the free boundary in the obstacle problem for the fractional Laplacian with drift at the critical order $s=1/2$, including general nonlocal operators.

Contribution

It establishes the regularity and precise expansion of the free boundary for the obstacle problem with critical fractional Laplacian and extends results to more general nonlocal operators.

Findings

Free boundary is $C^{1,eta}$ regular around regular points.

Explicit expansion of the solution near free boundary points.

Extension of results to general nonlocal operators of order 1.

Abstract

We study the obstacle problem for the fractional Laplacian with drift, $\min\left\{(-\Delta)^s u + b \cdot \nabla u,\,u -\varphi\right\} = 0$ 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 \[ u(x)-\varphi(x) = c_0\big((x-x_0)\cdot e\big)_+^{1+\tilde\gamma(x_0)} + o\left(|x-x_0|^{1+\tilde\gamma(x_0)+\sigma}\right), \] \[ \tilde{\gamma}(x_0) = \frac{1}{2}+\frac{1}{\pi} \arctan (b\cdot e), \] 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.