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 $$ \int_0^1-\log\left(1-\frac{\left|\int_0^xf(y) dy\right|^{p'}}{x}\right) dx\leq\mathsf{K}_p, \qquad \|f\|_{L^p(0,1)}\leq1, $$ where $\mathsf{K}_p$ 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 $\mathcal{M}_D(u)=|D|^{-1}\int_De^{4πu^2} dx$, we obtain $$ \mathcal{M}_D(u_*)-\mathcal{M}_D(u) \geq c \min_{\pm}\|\nabla(u\mp u_*)\|_{L^2(D)}^2 $$ for all $u\in H_0^1(D)$ with $\|\nabla u\|_{L^2(D)}\leq1$. Here $c>0$, and the exponent two is optimal, giving a quantitative stability theorem of Bianchi-Egnell type.