Necessary and sufficient conditions for rigidity of Pólya–Szegö inequality under Schwarz symmetrisation

Necessary and sufficient conditions for rigidity of Pólya–Szegö inequality under Schwarz symmetrisation

Pólya–Szegö inequality states that integral functionals, whose integrand is a convex function of the gradient, do not increase under Schwarz symmetrisation. In a seminal paper, Brothers and Ziemer gave sufficient conditions for the uniqueness (up to translations) of the extremal of the inequality. I...

Full description

Saved in:
Bibliographic Details
Published in:Nonlinear differential equations and applications Vol. 32; no. 1
Main Author: Cagnetti, Filippo
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 2025
Springer Nature B.V
Subjects:
Analysis
Mathematics
Mathematics and Statistics
Translations
Rearrangements of functions
Sobolev functions
Dirichlet integrals
Secondary 49Q10
Primary 49Q20
Pólya–Szegö inequality
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Pólya–Szegö inequality states that integral functionals, whose integrand is a convex function of the gradient, do not increase under Schwarz symmetrisation. In a seminal paper, Brothers and Ziemer gave sufficient conditions for the uniqueness (up to translations) of the extremal of the inequality. In this note we show that these conditions are also necessary.
ISSN:1021-9722
1420-9004
DOI:10.1007/s00030-024-01005-7