Speaker
Xavier Ros-Oton
Abstract
We study obstacle problems for parabolic operators of the type $\partial_t+L$, where $L$ is an elliptic integro-differential operator of order $2s$, in the critical and supercritical regimes $s=1/2$ and $s<1/2$. The only known result in this context was due to Caffarelli and Figalli, who established the $C^{1+s}_x$ regularity of solutions for the case $L = (−\Delta)^s$, the same regularity as in the elliptic setting.
Here we prove for the first time the regularity of free boundaries in both regimes, $s<1/2$ and $s=1/2$. Moreover, we also show that when $s<1/2$ solutions are actually more regular than in the elliptic case, and when $s=1/2$ we establish the optimal time regularity of solutions. Furthermore, our methods are very general, and allow us to extend all previous known results for $(-\Delta)^s$ to more general nonlocal operators $L$.
These are joint works with D. Torres-Latorre ($s<1/2$), and with A. Figalli and J. Serra ($s=1/2$).
Supported by the ERC under the Grant Agreement No 801867.