Fractional divergence-measure fields

The Gauss-Green and integration by parts formulas are of significant relevance in many areas of mathematical analysis and physics, and such applications motivated several investigations to extend these formulas to less regular integration domains and vector fields. These endeavours naturally led to the definition of the divergence-measure fields, which are L^p-summable vector fields whose divergence is a Radon measure. By applying a Leibniz rule between functions of bounded variation and essentially bounded divergence-measure fields, we will prove Gauss--Green formulas for these fields on sets with finite perimeter. It is also of interest to consider as integration domains sets with possibly fractal boundary, such as sets with finite fractional perimeter. To this purpose, we will present a distributional approach to fractional Sobolev spaces and fractional variation, which exploits the notions of fractional Riesz gradient and divergence. This will allow us introduce the fractional divergence-measure fields, which, in perfect analogy with the integer case, are L^p-summable vector fields whose fractional divergence is a Radon measure. Finally, we will provide Leibniz rules involving such fields and suitably regular scalar functions, leading to the fractional version of the Gauss-Green formula. The talk is mainly based on joint works with Kevin R. Payne and Giorgio Stefani.