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

TL;DR

This paper extends divergence-measure vector fields to fractional divergence settings, establishing properties like absolute continuity, Leibniz rule, and Gauss-Green formula, with sharpness discussed through examples.

Contribution

It introduces the space ^{,p}(b^n) for fractional divergence-measure fields and proves key properties extending classical results to this new setting.

Findings

Established absolute continuity of fractional divergence measures.

Derived fractional Leibniz rule and Gauss-Green formula.

Provided explicit examples demonstrating sharpness of results.

Abstract

Given $\alpha\in(0,1]$ and $p\in[1,+\infty]$, we define the space $\mathscr{DM}^{\alpha,p}(\mathbb R^n)$ 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.