Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Using epiperimetric inequalities approach, we study the obstacle problem $$ for the fractional Laplacian $$ with obstacle $$ , $$ and $$ . We prove an epiperimetric inequality for the Weiss' energy $$ and a logarithmic epiperimetric inequality for the Weiss' energy $$ . Moreover, we also prove two epiperimetric inequalities for negative energies $$ and $$ . By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies $$ and $$ . Finally, we give an alternative proof of the regularity of the points on the free boundary with frequency $$ and we describe the structure of the points on the free boundary with frequency 2m, with $$ and $$