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

Abstract

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. We localize our problem by considering a suitable extension operator introduced by L. Caffarelli and L. Silvestre. The structure of the extension equation is different from the one considered by L. Caffarelli, S. Salsa and L. Silvestre in their study of the obstacle problem for the fractional Laplacian without drift, 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 Almgren-type monotonicity formula, which we then use to establish the optimal regularity of solutions.