| Issue |
ESAIM: COCV
Volume 32, 2026
|
|
|---|---|---|
| Article Number | 55 | |
| Number of page(s) | 26 | |
| DOI | https://doi.org/10.1051/cocv/2026038 | |
| Published online | 07 July 2026 | |
On the optimal sets in Pólya and Makai type Inequalities
1
Mathematical and Physical Sciences for Advanced Materials and Technologies, Scuola Superiore Meridionale,
Largo San Marcellino 10,
80138 Napoli,
Italy
2
Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli studi di Napoli Federico II,
Via Cintia, Complesso Universitario Monte S. Angelo,
80126 Napoli,
Italy
3
Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague,
Trojanova 13,
120 00,
Prague,
Czech Republic
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
12
April
2025
Accepted:
6
May
2026
Abstract
In this paper, we examine some shape functionals, introduced by Pólya and Makai, involving the torsional rigidity and the first Dirichlet–Laplacian eigenvalue for bounded, open and convex sets of ℝn. We establish quantitative bounds, which give us key properties and information on the behavior of the optimizing sequences. In particular, we consider two kinds of remainder terms that provide information about the structure of these minimizing sequences, such as information about the thickness.
Mathematics Subject Classification: 35P15 / 49Q10 / 35J05 / 35J25
Key words: Pólya estimates / Makai estimates / quantitative inequalities / web functions
© The authors. Published by EDP Sciences, SMAI 2026
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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