On obstacle-type problems for higher order fractional Laplacian

In this talk, we will present our work on some one-phase and two-phase obstacle-type problems for higher order fractional Laplacian. As first observer by Yang, such a problem can be extended in the spirit of Caffarelli-Silvestre extension to a boundary obstacle-type problem in one extra dimension for a weighted bi-Laplace operator. Boundary obstacle problems governed by  operators of 4th order are in connection with unilateral phenomena for flat elastic plates. The study of the regularity of a solution is based on techniques from potential theory, while the analysis of the free boundary structure hinges on monotonicity formulas of Almgren- and Monneau-type. This work appear in two papers joint with Donatella Danielli and Arshak Petrosyan.