Segregated configurations involving the square root of the laplacian and their free boundaries

We study the local structure and the regularity of free boundaries of segregated minimal configurations involving the square root of the laplacian. We develop an improvement of flatness theory and, as a consequence of this and Almgren’s monotonicity formula, we obtain partial regularity (up to a small dimensional set) of the nodal set, thus extending the known results in Caffarelli and Lin (J Am Math Soc 21(3):847–862, 2008) and Tavares et al. (Calc Var Partial Differ Equ 45(3–4):273–317, 2012) for the standard diffusion to some anomalous case.