On the extremals of the P\'olya-Szeg\H{o} inequality

TL;DR

This paper investigates the characteristics of extremals in the Pólya-Szegő inequality, showing how their deviation from symmetric rearrangements relates to critical point measures.

Contribution

It provides a quantitative analysis linking extremals' deviation to the measure of critical points, advancing understanding of the inequality's extremal structure.

Findings

Deviation from symmetric rearrangement is controlled by critical point measure

Quantitative bounds are established for extremals

Insights into the structure of extremals in the inequality

Abstract

The distance of an extremal of the P\'olya-Szeg\H{o} inequality from a translate of its symmetric decreasing rearrangement is controlled by the measure of the set of critical points.