On the gradient rearrangement of functions

TL;DR

This paper introduces a symmetrization technique for BV functions' gradients, enabling comparison between original and symmetrized functions and deriving geometric inequalities.

Contribution

It presents a novel gradient symmetrization method for BV functions and applies it to establish Saint-Venant type inequalities for geometric functionals.

Findings

Established an L1 comparison between BV functions and their symmetrized versions

Separated the absolutely continuous and singular parts of the gradient in BV functions

Derived Saint-Venant type inequalities for specific geometric functionals

Abstract

In this paper, we introduce a symmetrization technique for the gradient of a $\BV$ function, which separates its absolutely continuous part from its singular part (sum of the jump and the Cantorian part). In particular, we prove an $\text{\emph{L}}^{\text{1}}$ comparison between the function and its symmetrized. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.