Some remarks on rearrangement for nonlocal functionals

Abstract

We prove that a nonlocal functional approximating the standard Dirichlet $p$-norm fails to decrease under two-point rearrangement. Furthermore, we get other properties related to this functional such as decay and compactness, and the Polya-Szegö inequality for Riesz fractional gradients, a notion recently introduced in the literature.

arXiv comment12 pages