Calculus of Variations and Geometric Measure Theory

G. E. Comi - V. De Cicco - G. Scilla

Beyond $BV$: new pairings and Gauss-Green formulas for measure fields with divergence measure

created by comi on 28 Oct 2023
modified on 20 Nov 2023



Inserted: 28 oct 2023
Last Updated: 20 nov 2023

Year: 2023


A new notion of pairing between measure vector fields with divergence measure and scalar functions, which are not required to be weakly differentiable, is introduced. In particular, in the case of essentially bounded divergence-measure fields, the functions may not be of bounded variation. This naturally leads to the definition of $BV$-like function classes on which these pairings are well defined. Despite the lack of fine properties for such functions, our pairings surprisingly preserve many features of the recently introduced $\lambda$-pairings (Crasta, De Cicco, Malusa), as coarea formula, lower semicontinuity, Leibniz rules, and Gauss-Green formulas. Moreover, in a natural way new anisotropic "degenerate" perimeters are defined, possibly allowing for sets with fractal boundary.

Keywords: functions of bounded variation, Gauss-Green formula, divergence-measure fields