Fractional obstacle problem

The obstacle problem is to seek a $s$-superharmonic function $u$ which lies above some smooth obstacle function $\phi$ in the interior of some domain $\Omega \subset \mathbb{R}^n$. Where $u > \phi$, $u$ is $s$-harmonic. The function satisfies Dirichlet conditions on $\mathbb{R}^n \setminus \Omega$, or one can require $|u|\rightarrow 0$ as $|x|\rightarrow \infty$ if $\Omega$ is, say, all of $\mathbb{R}^n$. The problem can be formulated as a variational problem as well, either through the extension or directly through a Dirichlet-like nonlocal energy on $\mathbb{R}^n$.

Solutions to the problem have optimal regularity in Holder class $C^{1,s}$. There is no native nondegeneracy to the problem, and so nondegeneracy conditions have to be imposed. About nonsingular free boundary points, the free boundary is a $C^{1,\alpha}$ surface of dimension $n-1$. The nature of a free boundary point is classified by the Almgren frequency formula.