Calculus of Variations and Geometric Measure Theory

G. E. Comi - G. Stefani

Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula

created by stefani on 28 Feb 2023
modified on 01 Jun 2024


Published Paper

Inserted: 28 feb 2023
Last Updated: 1 jun 2024

Journal: Boll. Unione Mat. Ital.
Volume: 17
Number: 2
Pages: 259–281
Year: 2024
Doi: 10.1007/s40574-023-00370-y

ArXiv: 2303.00834 PDF


Given $\alpha\in(0,1]$ and $p\in[1,+\infty]$, we define the space $\mathcal{DM}^{\alpha,p}(\mathbb R^n)$ of $L^p$ vector fields whose $\alpha$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $\alpha$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

Keywords: Hausdorff measure, Gauss-Green formula, fractional calculus, fractional divergence-measure fields, Leibniz rule