Optimal rearrangement problem and normalized obstacle problem in the fractional setting

Abstract

We consider an optimal rearrangement minimization problem involving the fractional Laplace operator $(-\Delta {)}^s$, $0<s<1$, and Gagliardo-Nirenberg seminorm $|u{|}_s$. 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 $$-(-\Delta {)}^{s}U-{\chi }_{\{U\le 0\}}\min \{-(-\Delta {)}^s{U}^{+};1\}={\chi }_{\{U>0\}},$$ which happens to be the fractional analogue of the normalized obstacle problem $\Delta u={\chi }_{\{u>0\}}$. A new section analyzing $s\to 1$ has been added.