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arXiv · 2303.00834

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

Giovanni E. Comi·Giorgio Stefani
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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.

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Giovanni E. Comi, Giorgio Stefani. 2023-03-01. Fractional divergence-measure fields, Leibniz rule and Gauss-Green formula. https://doi.org/10.1007/s40574-023-00370-y

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