Epiperimetric inequalities in the obstacle problem for the fractional Laplacian

Abstract

AbstractUsing epiperimetric inequalities approach, we study the obstacle problem $$\min \{(-\Delta )^su,u-\varphi \}=0,$$ min { ( - Δ ) s u , u - φ } = 0 , for the fractional Laplacian $$(-\Delta )^s$$ ( - Δ ) s with obstacle $$\varphi \in C^{k,\gamma }(\mathbb {R}^n)$$ φ ∈ C k , γ ( R n ) , $$k\ge 2$$ k ≥ 2 and $$\gamma \in (0,1)$$ γ ∈ ( 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 2 m . Moreover, we also prove two epiperimetric inequalities for negative energies $$W_{1+s}$$ W 1 + s and $$W_{2m}$$ W 2 m . By these epiperimetric inequalities, we deduce a frequency gap and a characterization of the blow-ups for the frequencies $$\lambda =1+s$$ λ = 1 + s and $$\lambda =2m$$ λ = 2 m . 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 ∈ N and $$2\,m\le k.$$ 2 m ≤ k .