Obstacle Problems

In this chapter, we study obstacle problems for nonlocal elliptic operators. In order to do so, we first prove the boundary Harnack inequality for such operators in Lipschitz domains, a crucial tool in the regularity theory for free boundary problems. After that, we establish the smoothness of free boundaries near regular points and prove the optimal regularity of solutions. We finish the chapter with a brief discussion of some further results and open problems.