Fractional optimal maximization problem and the unstable fractional obstacle problem

Abstract

We consider an optimal rearrangement maximization problem involving the fractional Laplace operator $(-\Delta {)}^s$, $0<s<1$, 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 $\alpha >0$ $$(-\Delta {)}^{s}u={\chi }_{\{u>\alpha \}}.$$

10 pages. Submitted