Optimal rearrangement problem and normalized obstacle problem in the fractional setting

We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Delta)(s), 0 < s < 1, and the Gagliardo seminorm vertical bar u vertical bar(s). We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satisfies -(Delta)U-s - chi({U <= 0}) min {-(-Delta)(s) U+; 1} = chi({U>0}), which happens to be the fractional analogue of the normalized obstacle problem Delta u = chi({u>0}).