Calculus of Variations and Geometric Measure Theory

G. E. Comi - G. Stefani

Leibniz rules and Gauss-Green formulas in distributional fractional spaces

created by comi on 26 Nov 2021
modified on 19 May 2022


Published Paper

Inserted: 26 nov 2021
Last Updated: 19 may 2022

Journal: J. Math. Anal. Appl.
Volume: 514
Number: 2
Pages: Paper No. 126312
Year: 2022
Doi: 10.1016/j.jmaa.2022.126312

ArXiv: 2111.13942 PDF


We apply the results established in fractional variation to prove some new fractional Leibniz rules involving $BV^{\alpha,p}$ and $S^{\alpha,p}$ functions, following the distributional approach adopted in the previous works blow-up, asymptotics I and asymptotics II. In order to achieve our main results, we revise the elementary properties of the fractional operators involved in the framework of Besov spaces and we rephraze the Kenig-Ponce-Vega Leibniz-type rule in our fractional context. We apply our results to prove the well-posedness of the boundary-value problem for a general $2\alpha$-order fractional elliptic operator in divergence form.

Keywords: Fractional Gradient, fractional divergence, fractional variation, Besov space, fractional Leibniz rule, Kenig-Ponce-Vega inequality, fractional Gauss-Green formula