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 critical 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 resultsfor the standard diffusion to some anomalous case.