Fractional optimal maximization problem and the unstable fractional obstacle problem

Abstract We consider an optimal rearrangement maximization problem involving the fractional Laplace operator ( − Δ ) s , 0 s 1 , and the Gagliardo-Nirenberg seminorm [ u ] s . We prove the existence of a maximizer, analyze its properties and show that it satisfies the unstable fractional obstacle problem equation for some α > 0 ( − Δ ) s u = χ { u > α } .