Optimal rearrangement problem and normalized obstacle problem in the fractional setting - Bonder, Julián Fernández; Cheng, Zhiwei; Mikayelyan, Hayk (2020)
We consider an optimal rearrangement minimization problem involving the fractional Laplace operator (−∆) s , 0 < s < 1, and the Gagliardo 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 satis es −(−∆) sU − χU≤0 min{−(−∆) sU + ; 1} = χ{U>0} , which happens to be the fractional analogue of the normalized obstacle problem ∆u = χ{u>0} .