Calculus of Variations and Geometric Measure Theory
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G. Mancini - L. Martinazzi

The Moser-Trudinger inequality and its extremals on a disk via energy estimates

created by martinazz on 17 Jul 2018

[BibTeX]

preprint

Inserted: 17 jul 2018

Year: 2016

ArXiv: 1608.07169 PDF

Abstract:

We study the Dirichlet energy of non-negative radially symmetric critical points $u_\mu$ of the Moser-Trudinger inequality on the unit disc in $\mathbb{R}^2$, and prove that it expands as $$4\pi+\frac{4\pi}{\mu{4}}+o(\mu{-4})\le \int{B1}
\nabla
u\mu
2dx\le 4\pi+\frac{6\pi}{\mu{4}}+o(\mu{-4}),\quad \text{as }\mu\to\infty,$$ where $\mu=u_\mu(0)$ is the maximum of $u_\mu$. As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical regime (only in the case of the unit disk). Our results are stable under sufficiently weak perturbations of the Moser-Trudinger functional. We explicitly identify the critical level of perturbation for which, although the perturbed Moser-Trudinger inequality still holds, the energy of its critical points converges to $4\pi$ from below. We expect, in some of these cases, that the existence of extremals does not hold, nor the existence of critical points in the supercritical regime.

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