# Optimal rearrangement problem and normalized obstacle problem in the fractional setting

## TL;DR

This paper studies an optimal rearrangement problem involving the fractional Laplacian, proving existence and uniqueness of solutions, analyzing their properties, and deriving a non-local PDE as a fractional analogue of the obstacle problem.

## Contribution

It introduces a fractional setting for the normalized obstacle problem, establishing existence, uniqueness, and properties of the minimizer, and explores the limit as s approaches 1.

## Key findings

- Existence and uniqueness of the minimizer for the fractional rearrangement problem.
- Derivation of a non-local PDE as a fractional analogue of the obstacle problem.
- Analysis of the behavior as the fractional parameter s approaches 1.

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