A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
Abstract
For p>1, with p'=p/(p−1), we prove the sharp dual inequality ∫01−log(1−(|∫0xf(y) dy|p')/x) dx≤Kp, |f|Lp(0,1)≤1, where Kp is the logarithm of the optimal value in Moser's one-dimensional formulation of the Moser-Trudinger inequality. The dual functional follows from the Gibbs variational principle, monotone transport and the Legendre transform. Its supremum is attained, giving a dual proof of the existence theorems of Carleson-Chang (1986) and Hudson-Leckband (2002); the classical concentration threshold is realized by equality cases in Hoelder's inequality. For p=2 we prove, with rigorous computer-assisted estimates, that the nonnegative dual maximizer is unique and nondegenerate. Hence the extremal of the Moser-Trudinger inequality on a disk D is unique up to sign, confirming conjectures of Adimurthi-Karthik-Giacomoni (2016) and of Chen-Lu-Xue-Zhu (2026). The positive extremal u* is nondegenerate, and, writing ℳD(u)=|D|−1∫De4πu2 dx, we obtain ℳD(u*)−ℳD(u) ≥ c min±|∇(u∓ u*)|L2(D)2 for all u∈ H01(D) with |∇ u|L2(D)≤1. Here c>0, and the exponent two is optimal, giving a quantitative stability theorem of Bianchi-Egnell type.
Related
- Uniqueness and comparison principle for Trudinger's parabolic equation with Hölder continuous boundary values
- Extremal Trudinger-Moser inequality associated with many point vortices with multi-intensities
- Trudinger-Moser type inequality in fractional Sobolev space with singularity on smooth submanifold
- The Trudinger inequality is true for M1,s spaces
- Sharp concentration levels and extremal functions for a planar Choquard-type Trudinger--Moser inequality
BibTeX
@misc{li2026dual,
title = {{A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals}},
author = {Jungang Li and Wen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25375},
eprint = {2610.25375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25375},
}
RIS
TY - UNPB
TI - A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals
AU - Jungang Li
AU - Wen Sun
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25375
UR - https://arxiv.org/abs/2610.25375
AB - For p>1, with p'=p/(p−1), we prove the sharp dual inequality ∫_0^1−log(1−(|∫_0^xf(y) dy|^p')/x) dx≤K_p, |f|_L^p(0,1)≤1, where K_p is the logarithm of the optimal value in Moser's one-dimensional formulation of the Moser-Trudinger inequality. The dual functional follows from the Gibbs variational principle, monotone transport and the Legendre transform. Its supremum is attained, giving a dual proof of the existence theorems of Carleson-Chang (1986) and Hudson-Leckband (2002); the classical concentration threshold is realized by equality cases in Hoelder's inequality. For p=2 we prove, with rigorous computer-assisted estimates, that the nonnegative dual maximizer is unique and nondegenerate. Hence the extremal of the Moser-Trudinger inequality on a disk D is unique up to sign, confirming conjectures of Adimurthi-Karthik-Giacomoni (2016) and of Chen-Lu-Xue-Zhu (2026). The positive extremal u_* is nondegenerate, and, writing ℳ_D(u)=|D|^−1∫_De^4πu^2 dx, we obtain ℳ_D(u_*)−ℳ_D(u) ≥ c min_±|∇(u∓ u_*)|_L^2(D)^2 for all u∈ H_0^1(D) with |∇ u|_L^2(D)≤1. Here c>0, and the exponent two is optimal, giving a quantitative stability theorem of Bianchi-Egnell type.
ER -