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

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