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 2, with and