On monotonicity of some functionals with variable exponent under symmetrization

TL;DR

This paper establishes necessary and sufficient conditions for a Pólya–Szegö type inequality involving variable exponent functionals, advancing understanding of symmetrization effects in variable exponent spaces.

Contribution

It provides a complete characterization of when the Pólya–Szegö inequality holds for functionals with variable exponents, filling a gap in the theory.

Findings

Identifies conditions for the inequality to hold

Clarifies the role of monotonicity in variable exponent functionals

Extends classical symmetrization results to variable exponent settings

Abstract

We give necessary and sufficient conditions for the P\'olya--Szeg\"o type inequality with variable exponent of summability.