Calculus of Variations and Geometric Measure Theory
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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 13 Oct 2021

[BibTeX]

Preprint

Inserted: 30 sep 2021
Last Updated: 13 oct 2021

Year: 2021

ArXiv: 2109.15263 PDF

Abstract:

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


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