Fractional optimal maximization problem and the unstable fractional obstacle problem

TL;DR

This paper investigates a maximization problem involving the fractional Laplace operator and Gagliardo-Nirenberg seminorm, establishing existence, properties, and linking it to an unstable fractional obstacle problem equation.

Contribution

It introduces a new optimal rearrangement maximization problem with fractional operators and proves the existence and properties of its maximizer, connecting it to an unstable fractional obstacle problem.

Findings

Existence of a maximizer for the problem.

The maximizer satisfies an unstable fractional obstacle equation.

Characterization of the maximizer's properties.

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 $\alpha>0$ $$(-\Delta)^s u=\chi_{\{u>\alpha\}}.$$