Epiperimetric inequalities in the obstacle problem for the fractional Laplacian.

Using epiperimetric inequalities approach, we study the obstacle problem min { ( - Δ ) s u , u - φ } = 0 , for the fractional Laplacian ( - Δ ) s with obstacle φ ∈ C k , γ ( R n ) , k ≥ 2 and γ ∈ ( 0 , 1 ) . We prove an epiperimetric inequality for the Weiss' energy W 1 + s and a logarithmic epiperimetric inequality for the Weiss' energy W 2 m . Moreover, we also prove two epiperimetric inequalities for negative energies W 1 + s and W 2 m . By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies λ = 1 + s and λ = 2 m . 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 2 m , with m ∈ N and 2 m ≤ k .