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Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian (2412.16066v1)

Published 20 Dec 2024 in math.AP

Abstract: We establish generic regularity results of free boundaries for solutions of the obstacle problem for the fractional Laplacian $(-\Delta)s$. 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.

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