Extremals for Trudinger-Moser inequalities via polygonal profiles
Abstract
The aim of this paper is to establish the existence of extremal functions for the Trudinger-Moser inequality with logarithmic weights (−log |x|)β in two dimensions throughout the full range 0≤ β<1. We also extend recent results on the existence of local maximizers for the Trudinger-Moser functional beyond the critical threshold. Our approach uses polygonal profiles to establish a lower bound uniform in the weight parameter. An explicit thirty-node profile gives the lower bound TM4π>4.38421793217 π for the classical critical supremum on the unit disk, improving on the value 4.3556 π obtained in the computer experiments of Carleson and Chang. This lower bound has a purely analytic proof, which uses only elementary inequalities and exact rational arithmetic. Numerical experiments with polygonal profiles of up to $102400$ nodes suggest that this lower bound can be raised to 4.402732044 π, which improves the recent computational lower bound by Li and Sun.
Related
- Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
- Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
- A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
- The compactness of Moser-Trudinger type inequalities in the unit ball
- Semiclassical Moser--Trudinger type inequalities for orthonormal families in ℝ2
BibTeX
@misc{oliveira2026extremals,
title = {{Extremals for Trudinger-Moser inequalities via polygonal profiles}},
author = {José Francisco de Oliveira},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33439},
eprint = {2610.33439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33439},
}
RIS
TY - UNPB
TI - Extremals for Trudinger-Moser inequalities via polygonal profiles
AU - José Francisco de Oliveira
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.33439
UR - https://arxiv.org/abs/2610.33439
AB - The aim of this paper is to establish the existence of extremal functions for the Trudinger-Moser inequality with logarithmic weights (−log |x|)^β in two dimensions throughout the full range 0≤ β<1. We also extend recent results on the existence of local maximizers for the Trudinger-Moser functional beyond the critical threshold. Our approach uses polygonal profiles to establish a lower bound uniform in the weight parameter. An explicit thirty-node profile gives the lower bound TM_4π>4.38421793217 π for the classical critical supremum on the unit disk, improving on the value 4.3556 π obtained in the computer experiments of Carleson and Chang. This lower bound has a purely analytic proof, which uses only elementary inequalities and exact rational arithmetic. Numerical experiments with polygonal profiles of up to $102400$ nodes suggest that this lower bound can be raised to 4.402732044 π, which improves the recent computational lower bound by Li and Sun.
ER -