A dual variational principle for the Moser-Trudinger inequality: existence, uniqueness and stability of extremals

For $p>1$, with $p^{\mathrm{\prime }}=p\mathrm{/}(p-1)$, we prove the sharp dual inequality $${\int }_0^1-\log (1-\frac{{\mid {\int }_0^{x}f(y)dy\mid }^{p^{\mathrm{\prime }}}}{x})dx\le {\mathsf{K}}_p,\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(0,1)}\le 1,$$ 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 $u_{\ast }$ is nondegenerate, and, writing ${\mathcal{M}}_D(u)=\mathrm{\mid }D{\mathrm{\mid }}^{-1}{\int }_D{e}^{4\pi u^2}dx$, we obtain $${\mathcal{M}}_D(u_{\ast })-{\mathcal{M}}_D(u)\ge c\underset{\pm }{\min }\mathrm{\parallel }\mathrm{\nabla }(u\mp u_{\ast }){\mathrm{\parallel }}_{L^2(D)}^2$$ for all $u\in H_0^1(D)$ with $\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2(D)}\le 1$. Here $c>0$, and the exponent two is optimal, giving a quantitative stability theorem of Bianchi-Egnell type.