On monotonicity of some functionals with variable exponent under symmetrization

ABSTRACT

We give necessary and sufficient conditions for the Pólya–Szegö-type inequality with variable exponent of summability.

KEYWORDS:

• Symmetrization
• variable exponent
• Pólya–Szegö inequality

AMS SUBJECT CLASSIFICATIONS:

• 49K05
• 49N99

Acknowledgements

We are grateful to M. Surnachev for simplification of the proof of Theorem 3.3 and to an anonymous referee for important advices.

Notes

No potential conflict of interest was reported by the authors.

To the memory of V.V. Zhikov.

Additional information

Funding

The paper is supported by RFBR [grant number 18-01-00472].