We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like ${(-\mathrm{\Delta })}^s+b\cdot \mathrm{\nabla }$, in the subcritical regime $s>\frac{1}{2}$. Our main result states that once the free boundary is $C^1$ then it is $C^{\infty }$, whenever $s\notin \mathbb{Q}$.

In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that s is irrational plays al important role. Such expansion still allows us to prove a higher order boundary Harnack inequality, where the regularity holds in the tangential directions only.

我们研究了具有漂移的积分微分算子的障碍问题中自由边界的更高规律性，例如 ${(-\mathrm{\Delta })}^{秒}+乙\cdot \mathrm{\nabla }$, 在亚临界状态 $秒>\frac{1}{2}$. 我们的主要结果表明，一旦自由边界是$C^1$ 那么它是 $C^{\infty }$，每当 $秒\notin \mathbb{问}$.

为了实现这一点，我们为具有距离函数幂的漂移的线性非局部方程的解建立了精细的边界扩展。非常有趣的是，由于漂移项，幂不会按自然数增加，而s是无理数这一事实起着重要作用。这种扩展仍然允许我们证明高阶边界 Harnack 不等式，其中规律性仅在切线方向成立。