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

Abstract

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.

arXiv comment22 pages

References in corpus (12)

• A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up
• Anzellotti's pairing theory and the Gauss--Green theorem
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I
• A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
• The Dirichlet problem for singular elliptic equations with general nonlinearities
• Cauchy Fluxes and Gauss-Green Formulas for Divergence-Measure Fields over General Open Sets
• An extension of the pairing theory between divergence-measure fields and BV functions
• The Dirichlet problem for the $1$-Laplacian with a general singular term and $L^1$-data
• Rigidity and trace properties of divergence-measure vector fields
• The fractional variation and the precise representative of $B{V}^{\alpha ,p}$ functions
• Leibniz rules and Gauss-Green formulas in distributional fractional spaces
• Failure of the local chain rule for the fractional variation