Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Using the epiperimetric inequalities approach, we study the obstacle problem $\min\{(-\Delta)^su,u-\varphi\}=0,$ for the fractional Laplacian $(-\Delta)^s$ with obstacle $\varphi\in C^{k,\gamma}(\mathbb{R}^n)$, $k\ge2$ and $\gamma\in(0,1)$. We prove an epiperimetric inequality for the Weiss' energy $W_{1+s}$ and a logarithmic epiperimetric inequality for the Weiss' energy $W_{2m}$. Moreover, we also prove two epiperimetric inequalities for negative energies $W_{1+s}$ and $W_{2m}$. By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies $\lambda=1+s$ and $\lambda=2m$. Finally, we give an alternative proof of the regularity of the points on the free boundary with frequency $1+s$ and we describe the structure of the points on the free boundary with frequency $2m$, with $m\in\mathbb{N}$ and $2m\le k.$