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arXiv · 2412.16066

Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian

Matteo Carducci·Roberto Colombo
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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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Matteo Carducci, Roberto Colombo. 2024-12-20. Generic regularity of free boundaries in the obstacle problem for the fractional Laplacian. https://arxiv.org/abs/2412.16066

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