Calculus of Variations and Geometric Measure Theory

G. E. Comi - G. Stefani

Failure of the local chain rule for the fractional variation

created by comi on 19 May 2022
modified by stefani on 27 Jan 2023


Accepted Paper

Inserted: 19 may 2022
Last Updated: 27 jan 2023

Journal: Port. Math.
Year: 2022

ArXiv: 2206.03197 PDF


We prove that the local version of the chain rule cannot hold for the fractional variation defined in blow-up. In the case $n = 1$, we prove a stronger result, exhibiting a function $f \in BV^{\alpha}(\mathbb{R})$ such that $\vert f\vert \notin BV^{\alpha}(\mathbb{R})$. The failure of the local chain rule is a consequence of some surprising rigidity properties for non-negative functions with bounded fractional variation which, in turn, are derived from a fractional Hardy inequality localized to half-spaces. Our approach exploits the results of Leibniz rules and the distributional approach of the previous papers blow-up, asymptotics I, asymptotics II, fractional variation. As a byproduct, we refine the fractional Hardy inequality obtained in Shieh-Spector II, Spector and we prove a fractional version of the closely related Meyers-Ziemer trace inequality.

Keywords: Chain Rule, Fractional Gradient, fractional divergence, fractional variation, fractional Hardy inequality