Calculus of Variations and Geometric Measure Theory

G. E. Comi - D. Spector - G. Stefani

The fractional variation and the precise representative of $BV^{\alpha,p}$ functions

created by comi on 30 Sep 2021
modified on 19 May 2022


Published Paper

Inserted: 30 sep 2021
Last Updated: 19 may 2022

Journal: Fract. Calc. Appl. Anal.
Volume: 25
Pages: 520-558
Year: 2022
Doi: 10.1007/s13540-022-00036-0

ArXiv: 2109.15263 PDF


We continue the study of the fractional variation following the distributional approach developed in the previous works blow-up, asymptotics I and asymptotics II. We provide a general analysis of the distributional space $BV^{\alpha,p}(\mathbb{R}^n)$ of $L^p$ functions, with $p\in[1,+\infty]$, possessing finite fractional variation of order $\alpha\in(0,1)$. Our two main results deal with the absolute continuity property of the fractional variation with respect to the Hausdorff measure and the existence of the precise representative of a $BV^{\alpha,p}$ function.

Keywords: Hausdorff measure, Fractional Gradient, fractional divergence, fractional variation, fractional capacity, precise representative