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

Abstract

AbstractGiven$$\alpha \in (0,1]$$α∈(0,1]and$$p\in [1,+\infty ]$$p∈[1,+∞], we define the space$${\mathcal {D}}{\mathcal {M}}^{\alpha ,p}({\mathbb {R}}^n)$$DMα,p(Rn)of$$L^p$$Lpvector fields whose$$\alpha $$α-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$$\alpha $$α-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.