Improved Moser-Trudinger inequality of Tintarev type in dimension \(n\) and the existence of its extremal functions
type: Item
instance of: scholarly article
title: Improved Moser-Trudinger inequality of Tintarev type in dimension \(n\) and the existence of its extremal functions
zbMATH Open document ID: 1405.26019
published in: Annals of Global Analysis and Geometry
publication date: 2018-11-01T00:00:00Z
review text: For a smooth bounded domain \(\Omega\subset \mathbb{R}^n\) with \(n\geq 2\), let \(W_0^{1,n} (\Omega)\) be the usual Sobolev space on \(\Omega\) and define \(\lambda_1 (\Omega) = \inf_{u\in W_0^{1,n} (\Omega)\setminus \{0\}} \frac{\int_{\Omega} |\nabla u |^n dx }{\int_{\Omega} |u|^n dx}\). Based on the blow-up analysis method, the author proves the following improved Moser-Trudinger inequality of Tintarev type \[ \sup_{u\in W_0^{1,n} , \int_{\Omega} |\nabla u |^n dx - \alpha \int_{\Omega} |u|^n dx \leq 1} \int_{\Omega} \exp (\alpha_n |u|^{\frac{n}{n-1}} ) dx < \infty\tag{1} \] for any \(0 \leq \alpha < \lambda_1 (\Omega)\), where \(\alpha_n = n \omega_{n-1}^{\frac{1}{n-1}}\) for the surface area \(\omega_{n-1}\) of the unit sphere in \(\mathbb{R}^n\). The inequality (1) is stronger than the improved Moser-Trudinger inequality obtained by \textit{Adimurthi} and \textit{O. Druet} [Commun. Partial Differ. Equations 29, No. 1--2, 295--322 (2004; Zbl 1076.46022)] in dimension 2 and by \textit{Y. Yang} [J. Funct. Anal. 239, No. 1, 100--126 (2006; Zbl 1141.46015)] in higher dimension and extends a result of \textit{C. Tintarev} [J. Funct. Anal. 266, No. 1, 55--66 (2014; Zbl 1303.26022)] in dimension 2 to higher dimension. Furthermore, the author proves that the supremun in (1) is attained for any \(0< \alpha < \lambda_1 (\Omega)\). The paper is systematically well-organized.
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 6970066
zbMATH Keywords: improved Moser-Trudinger inequality
zbMATH Keywords: blow-up analysis
zbMATH Keywords: extremal function
zbMATH Keywords: elliptic regularity theory
author: Nguyen Van Hoang
reviewed by: Choonkil Park
MaRDI profile type: Publication
full work available at URL: s10455-018-9599-z
OpenAlex ID: W2790361892
cites work: A sharp inequality of J. Moser for higher order derivatives
cites work: Blow-up Analysis in Dimension 2 and a Sharp Form of Trudinger–Moser Inequality
cites work: Q3581209
cites work: Fundamental solution for the \(Q\)-Laplacian and sharp Moser--Trudinger inequality in Carnot groups.
cites work: Q3756181
cites work: Concentration-compactness principles for Moser-Trudinger inequalities: new results and proofs
cites work: Conformal deformations of metric on \({\mathcal S}\) 2
cites work: Moser-Trudinger inequalities without boundary conditions and isoperimetric problems
cites work: Best constants for Moser-Trudinger inequalities on the Heisenberg group
cites work: Extremal functions for the singular Moser-Trudinger inequality in 2 dimensions
cites work: Singular Moser–Trudinger inequality on simply connected domains
cites work: On an inequality by N. Trudinger and J. Moser and related elliptic equations
cites work: A sharp inequality of Trudinger-Moser type and extremal functions in \(H^{1, n}(\mathbb{R}^n)\)
cites work: Trudinger–Moser inequality on the whole plane and extremal functions
cites work: Q4827374
cites work: A classification result for the quasi-linear Liouville equation
cites work: Extremal functions for the Trudinger-Moser inequality in 2 dimensions
cites work: Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications
cites work: A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument
cites work: Moser-Trudinger inequality on compact Riemannian manifolds of dimension two
cites work: Extremal functions for the Moser-Trudinger inequalities on compact Riemannian manifolds
cites work: A sharp Trudinger-Moser type inequality for unbounded domains in $\mathbb{R}^n$
cites work: Extremal functions for Moser’s inequality
cites work: The concentration-compactness principle in the calculus of variations. The limit case. I
cites work: MOSER–TRUDINGER INEQUALITY ON CONFORMAL DISCS
cites work: Trudinger-Moser inequality in the hyperbolic space \(\mathbb H^N\)
cites work: Q5614914
cites work: Improved Moser-Trudinger inequality for functions with mean value zero in \(\mathbb{R}^n\) and its extremal functions
cites work: Q5516356
cites work: A sharp Trudinger-Moser type inequality for unbounded domains in \(\mathbb R^2\)
cites work: Local behavior of solutions of quasi-linear equations
cites work: Critical points of embeddings of \(H_ 0^{1,n}\) into Orlicz spaces
cites work: Trudinger-Moser inequality with remainder terms
cites work: Q5547919
cites work: Regularity for a more general class of quasilinear equations
cites work: A sharp form of Moser-Trudinger inequality in high dimension
cites work: EXTREMAL FUNCTIONS FOR A SHARP MOSER–TRUDINGER INEQUALITY
cites work: A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface
cites work: Trudinger-Moser inequalities on complete noncompact Riemannian manifolds
cites work: Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two
cites work: Sharp Moser-Trudinger inequalities on Riemannian manifolds with negative curvature
cites work: Blow-up analysis concerning singular Trudinger-Moser inequalities in dimension two
cites work: An improved singular Trudinger-Moser inequality in unit ball
cites work: Q5520312
DOI: 10.1007/S10455-018-9599-Z
description: scientific article; zbMATH DE number 6970066