Obstacle problems for integro-differential operators: regularity of solutions and free boundaries

摘要

We study the obstacle problem for integro-differential operators of order 2s, with s is an element of(0,1). Our main results establish the optimal C<^>{1,s} regularity of solutions u, and the C<^>{1,lpha} regularity of the free boundary near regular points. Namely, we prove the following dichotomy at all free boundary points x_0inpartial{u=arphi}: