Obstacle problems for higher-order fractional powers of the Laplacian

In this talk we will discuss a class of boundary obstacle problems associated with the fractional Laplacian $(−\Delta)^s$ , for $1 < s < 2$. Our goals are to establish regularity properties of the solution and to describe the structure of the free boundary. To this end, we combine classical techniques from PDEs and the calculus of variations with more modern methods, such as the localization of the operator and monotonicity formulas. This is joint work with A. Haj Ali (University of Michigan) and G. Gravina (Loyola University - Chicago).