Anisotropic gradient rearrangement of BV functions and applications

TL;DR

This paper develops an anisotropic symmetrization technique for BV functions, extending Euclidean results, and applies it to derive isoperimetric inequalities for geometric functionals like torsional rigidity.

Contribution

It introduces a novel anisotropic symmetrization method for BV functions, generalizing previous Euclidean results and enabling new geometric inequalities.

Findings

Established an $L^1$ comparison between functions and their anisotropic symmetrizations.

Derived isoperimetric inequalities for torsional rigidity and related functionals.

Separated absolutely continuous and singular parts of the anisotropic gradient.

Abstract

In this paper, we introduce a symmetrization technique for the distributional gradient of a function of bounded variation in the anisotropic setting. This generalizes the result obtained in the Euclidean case in [Amato-Gentile-Nitsch-Trombetti, 2024] by separating the absolutely continuous part of the anisotropic gradient from its singular part. Our main result is an $L^1$ comparison between the function and its anisotropic symmetrization. Moreover, as an application, we derive isoperimetric inequalities for some geometric functionals related to the torsional rigidity.