Optimal rearrangement problem and normalized obstacle problem in the fractional setting

Abstract

We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (-Δ)s, 0 < s < 1, and the Gagliardo seminorm jujs. We prove the existence of the unique minimizer, analyze its properties as well as derive the non-local and highly non-linear PDE it satises -(-Δ)sU - x-(-Δ)sU+; 1 =U>0g, which happens to be the fractional analogue of the normalized obstacle problem Δu = xu>0.