2016/10/13 by Fernández-Real, Xavier, Ros-Oton, Xavier
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1610.04200
We study the obstacle problem for the fractional Laplacian with drift, min\(-Δ)s u + b ⋅ ∇ u, u -φ\ = 0 in ℝn, in the critical regime s = (1)/(2). Our main result establishes the C1,α regularity of the free boundary around any regular point x0, with an expansion of the form u(x)-φ(x) = c0((x-x0)⋅ e)+1+γ(x0) + o(|x-x0|1+γ(x0)+σ), γ(x0) = (1)/(2)+\frac1π \arctan (b⋅ e), where e ∈ \mathbbSn-1 is the normal vector to the free boundary, σ>0, and c0> 0. We also establish an analogous result for more general nonlocal operators of order 1. In this case, the exponent γ(x0) also depends on the operator.