Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

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