Short Talk: Xiangqian Meng

In this talk, we consider the obstacle problems for nonlocal operators with stable-like jump kernels $b(x,y)$, where $x$ and $y$ are the starting the landing positions and $b(x,y)$ satisfies the bounded, symmetric, Holder continuous conditions. Under the condition that the obstacle function $\varphi\in C_c^{\infty}(\R^d)$, we show the existence and uniqueness of the solution. We prove the regularity of the solution, which extends the work by Caffarelli, Ros-Oton and Serra in 2008.We conjecture that the free boundary near the regular point which the density near those points are thick enough is at least locally Lipschitz.This is an ongoing project.