Calculus of Variations and Geometric Measure Theory

G. E. Comi - G. P. Leonardi

Measures in the dual of $BV$: perimeter bounds and relations with divergence-measure fields

created by comi on 06 Jul 2024



Inserted: 6 jul 2024
Last Updated: 6 jul 2024

Pages: 41
Year: 2024


We analyze some properties of the measures in the dual of the space $BV$, by considering (signed) Radon measures satisfying a perimeter bound condition, which means that the absolute value of the measure of a set is controlled by the perimeter of the set itself, and whose total variations also belong to the dual of $BV$. We exploit and refine the results of (Phuc, Torres 2017), in particular exploring the relation with divergence-measure fields and proving the stability of the perimeter bound from sets to $BV$ functions under a suitable approximation of the given measure. As an important tool, we obtain a refinement of Anzellotti-Giaquinta approximation for $BV$ functions, which is of separate interest in itself and, in the context of Anzellotti's pairing theory for divergence-measure fields, implies a new way of approximating $\lambda$-pairings, as well as new bounds for their total variation. These results are also relevant due to their application in the study of weak solutions to the non-parametric prescribed mean curvature equation with measure data, which is explored in a subsequent work.

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