Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Abstract

Using the epiperimetric inequalities approach, we study the obstacle problem $\min \{(-\Delta {)}^{s}u,u-\phi \}=0,$ for the fractional Laplacian $(-\Delta {)}^s$ with obstacle $\phi \in C^{k,\gamma }({\mathbb{R}}^n)$, $k\ge 2$ 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 $2m\le k.$