论文标题
漂移的分数障碍问题：自由边界的较高规律性
The fractional obstacle problem with drift: higher regularity of free boundaries
论文作者
论文摘要
我们研究了具有漂移的障碍物问题中较高的自由边界的规律性，例如$（δ）^s +b \ cdot \ nabla $，在亚临界体制$ s> \ frac {1} {2} {2} $中。我们的主要结果指出，一旦自由边界为$ c^1 $，则每当$ s \ not \ in \ mathbb {q} $中，它是$ c^\ infty $。 为了实现这一目标，我们建立了良好的边界扩展，以在距离函数的能力方面对线性非局部方程进行漂移。有趣的是，由于漂移术语，这些权力不会随自然数量而增加，而$ s $是非理性的重要作用的事实。这样的扩展仍然使我们能够证明较高的边界不平等现象，而规则性仅在切向方向上。
We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-Δ)^s +b\cdot\nabla$, 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 $s\not\in\mathbb{Q}$. 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.
