Optimal regularity of solutions to the obstacle problem for the fractional Laplacian with drift

We prove existence, uniqueness and optimal regularity of solutions to the stationary obstacle problem defined by the fractional Laplacian operator with drift, in the subcritical regime. As in [4], we localize our problem by considering a suitable extension operator introduced in [2]. The structure of the extension equation is different from the one constructed in [4], in that the obstacle function has less regularity, and exhibits some singularities. To take into account the new features of the problem, we prove a new monotonicity formula, which we then use to establish the optimal regularity of solutions.