Global regularity for the free boundary in the obstacle problem for the fractional Laplacian

We study the regularity of the free boundary in the obstacle problem for the fractional Laplacian under the assumption that the obstacle $\varphi$ satisfies $\Delta\varphi\leq 0$ near the contact region. Our main result establishes that the free boundary consists of a set of regular points, which is known to be a $(n-1)$-dimensional $C^{1,\alpha}$ manifold by the results obtained by Caffarelli, Salsa, and Silvestre ({\it Invent. Math.} 2008), and a set of singular points, which we prove to be contained in a union of $k$-dimensional $C^1$-submanifold, $k=0,\ldots, n-1$. Such a complete result on the structure of the free boundary, proved by L. A. Caffarelli ({\it Acta Math.} 1977 and {\it J. Fourier Anal. Appl.} 1998), was known only in the case of the classical Laplacian, and it is new even for the Signorini problem (which corresponds to the particular case of the ${1\over 2}$-fractional Laplacian). A key ingredient behind our results is the validity of a new non-degeneracy condition $\sup_{B_r(x_0)}(u-\varphi)\geq c\,r^2$, valid at all free boundary points $x_0$.