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 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 $$ -(-\Delta)^s U-\chi_{\{U\leq 0\}}\min\{-(-\Delta)^s U^+;1\}=\chi_{\{U>0\}}, $$ 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.