The fractional obstacle problem with drift: higher regularity of free boundaries

TL;DR

This paper proves that free boundaries in obstacle problems for certain nonlocal operators with drift are infinitely smooth if initially $C^1$, especially when the fractional order is irrational, by developing a detailed boundary expansion and higher order inequalities.

Contribution

It introduces a novel boundary expansion for solutions with drift, showing higher regularity of free boundaries in nonlocal obstacle problems when the fractional order is irrational.

Findings

Free boundaries are $C^ abla$ if initially $C^1$ and $s$ is irrational.

Established a boundary expansion involving non-integer powers due to drift.

Proved a higher order boundary Harnack inequality in tangential directions.

Abstract

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like $(-\Delta)^s +b\cdot\nabla$, 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.