Duality and best constant for a Trudinger-Moser inequality involving probability measures
type: Item
instance of: scholarly article
title: Duality and best constant for a Trudinger-Moser inequality involving probability measures
zbMATH Open document ID: 1300.26017
DOI: 10.4171/JEMS/462
author: Tonia Ricciardi
author: Takashi Suzuki
published in: Journal of the European Mathematical Society (JEMS)
publication date: 2014-10-10T00:00:00Z
reviewed by: Teodora-Liliana Rădulescu
Mathematics Subject Classification ID: 26D15
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6353603
zbMATH Keywords: Trudinger-Moser inequality
zbMATH Keywords: mean field equation
zbMATH Keywords: logarithmic Hardy-Littlewood-Sobolev inequality
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/jems/462
OpenAlex ID: W2015904334
review text: Let \(\Omega\) be a two-dimensional Riemannian surface without boundary and assume that \(\lambda\) is a positive parameter. Denote \(I=[-1,1]\) and suppose that \(\mathcal P\) is a Borel probability measure on \(I\). For all \(v\in H^1(\Omega)\) satisfying \(\int_\Omega v=0\), consider the Trudinger-Moser energy functional NEWLINE\[NEWLINE J_\lambda(v)=\frac{1}{2}\int_\Omega|\nabla v|^2-\lambda\int_I\left(\log\int_\Omega e^{\alpha v}\right)\,\mathcal P(d\alpha).NEWLINE\]NEWLINE The main result establishes a Toland non-convex duality principle for \(J_\lambda\) and a computation of the optimal value of \(\lambda\) for which \(J_\lambda\) is bounded from below. The study of such problems is motivated by models arising in the statistical mechanics description of equilibrium turbulence, under the assumption that the intensity and the orientation of the vortices are determined by \(\mathcal P\).
description: scientific article; zbMATH DE number 6353603