# Structure and regularity of the singular set in the obstacle problem for the fractional Laplacian

## TL;DR

This paper investigates the structure and regularity of the singular set in the obstacle problem involving the fractional Laplacian, extending monotonicity formulas to all fractional orders between 0 and 1.

## Contribution

It establishes the complete structure and regularity of the singular set for the fractional obstacle problem, introducing new Monneau-type monotonicity formulas applicable to all s in (0,1).

## Key findings

- Complete characterization of the singular set structure.
- New monotonicity formulas for all fractional orders.
- Extension of previous results to general obstacles.

## Abstract

We study the singular part of the free boundary in the obstacle problem for the fractional Laplacian, \ $\min\bigl\{(-\Delta)^su,\,u-\varphi\bigr\}=0$ in $\mathbb R^n$, for general obstacles $\varphi$. Our main result establishes the complete structure and regularity of the singular set. To prove it, we construct new monotonicity formulas of Monneau-type that extend those in \cite{GP} to all $s\in(0,1)$.