Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian
Matteo Carducci, Roberto Colombo · arXiv (Cornell University) · 2024
We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-Δ)^s$. We prove that, for almost every obstacle, the free boundary contains only regular points up to dimension $3$, for every $s\in(0,1)$. To do so, we extend some results on the fine structure of the free boundary to the case $s\in (0,1)$ and general non-zero obstacle, including a blow-up analysis at points with frequency $2m+2s$, and we prove new explicit uniform frequency gaps for solutions of the fractional obstacle problem.