Mancini, Gabriele and Martinazzi, Luca.
(2016)
* The Moser-Trudinger inequality and its extremals on a disk via energy estimates.*
Preprints Fachbereich Mathematik, 2016 (17).

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## 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_{B_1} |\grad u_\mu|^2 dx \le 4\pi + \frac{6\pi}{\mu^4}+o(\mu^{-4}), 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 su ciently 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\pi4 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.

\[

4\pi + \frac{4\pi}{\mu^4}+o(\mu^{-4}) \le \int_{B_1} |\grad u_\mu|^2 dx \le 4\pi + \frac{6\pi}{\mu^4}+o(\mu^{-4}), 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 su ciently 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\pi4 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.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) 12 Special Collections > Preprints Fachbereich Mathematik |
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UniBasel Contributors: | Mancini, Gabriele and Martinazzi, Luca |

Item Type: | Preprint |

Publisher: | Universität Basel |

Language: | English |

Last Modified: | 06 May 2019 22:39 |

Deposited On: | 28 Mar 2019 09:51 |

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