edoc: No conditions. Results ordered -Date Deposited. 2024-10-08T16:17:38ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2020-09-09T08:28:43Z2020-09-09T08:31:38Zhttps://edoc.unibas.ch/id/eprint/59716This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/597162020-09-09T08:28:43ZThe Moser-Trudinger inequality and its extremals on a disk via energy estimates We study the Dirichlet energy of non-negative radially symmetric critical points uμ of the Moser–Trudinger inequality on the unit disc in R2, and prove that it expands as 4π+4πμ4+o(μ−4)≤∫B1|∇uμ|2dx≤4π+6πμ4+o(μ−4),as μ→∞, where μ=uμ(0) is the maximum of uμ. 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 4π 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. Gabriele ManciniLuca Martinazzi2019-03-28T09:51:46Z2019-05-06T22:39:10Zhttps://edoc.unibas.ch/id/eprint/69975This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699752019-03-28T09:51:46ZThe Moser-Trudinger inequality and its extremals on a disk via energy estimatesWe 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. Gabriele ManciniLuca Martinazzi2019-03-28T09:51:38Z2019-05-06T22:33:21Zhttps://edoc.unibas.ch/id/eprint/69957This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699572019-03-28T09:51:38ZExtremal functions for singular Moser-Trudinger embeddingsWe study Moser-Trudinger type functionals in the presence of singular potentials. In particular we propose a proof of a singular Carleson-Chang type estimate by means of Onofri’s inequality for the unit disk in $\mathbb{R}^2$. Moreover we extend the analysis of [1] and [8] considering Adimurthi-Druet type functionals on compact surfaces with conical singularities and discussing the existence of extremals for such functionals. Stefano IulaGabriele Mancini2017-11-30T12:34:09Z2017-11-30T12:34:09Zhttps://edoc.unibas.ch/id/eprint/43974This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/439742017-11-30T12:34:09ZRemarks on the Moser-Trudinger inequalityLuca BattagliaGabriele Mancini2017-11-02T10:19:32Z2017-11-02T10:19:32Zhttps://edoc.unibas.ch/id/eprint/43975This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/439752017-11-02T10:19:32ZA note on compactness properties of the singular Toda systemIn this note, we consider blow-up for solutions of the SU(3) Toda system on a compact surface Σ. In particular, we give a complete proof of the compactness result stated by Jost, Lin and Wang in [11] and we extend it to the case of singularities. This is a necessary tool to find solutions through variational methods. Luca BattagliaGabriele Mancini2017-11-02T09:39:50Z2017-11-02T09:39:50Zhttps://edoc.unibas.ch/id/eprint/43976This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/439762017-11-02T09:39:50ZOnofri-type inequalities for singular Liouville equationsWe study the blow-up behavior of minimizing sequences for the singular Moser–Trudinger functional on compact surfaces. Assuming non-existence of minimum points, we give an estimate for the infimum value of the functional. This result can be applied to give sharp Onofri-type inequalities on the sphere in the presence of at most two singularities. Gabriele Mancini