Anisotropic symmetrization, convex bodies, and isoperimetric inequalities

TL;DR

This paper establishes a new geometric approach to anisotropic symmetrization inequalities for Sobolev functions, avoiding approximation and enabling characterization of extremal functions.

Contribution

It introduces a novel proof method based on anisotropic isoperimetric inequalities and convex geometry, providing deeper geometric insights and direct extremal function characterization.

Findings

Proves a Pólya-Szegő type inequality for anisotropic functionals.

Develops a new approach using convex bodies and Brunn-Minkowski theory.

Characterizes extremal functions without approximation arguments.

Abstract

This work is concerned with a P\'olya-Szeg\"o type inequality for anisotropic functionals of Sobolev functions. The relevant inequality entails a double-symmetrization involving both trial functions and functionals. A new approach that uncovers geometric aspects of the inequality is proposed. It relies upon anisotropic isoperimetric inequalities, fine properties of Sobolev functions, and results from the Brunn-Minkowski theory of convex bodies. Importantly, unlike previously available proofs, the one offered in this paper does not require approximation arguments and hence allows for a characterization of extremal functions.