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

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like ( ? ? ) s + b ? ? , subcritical regime > 1 2 . Our main result states that once boundary is C then it ? whenever ? Q In order to achieve this, we establish a fine expansion solutions linear nonlocal equations drift terms powers distance function. Quite interestingly, due term, do not increase by natural numbers and fact irrational plays al important role. Such still allows us prove Harnack inequality, where holds tangential directions only.