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)su=χ{u>α}.