The affine Sobolev inequality.
type: Item
instance of: scholarly article
title: The affine Sobolev inequality.
zbMATH Open document ID: 1040.53089
author: Gaoyong Zhang
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
full work available at URL: 1 6.pdf
review text: The author proves a new Sobolev inequality which is stronger than its classical, Euclidean counterpart. The main theorem in the article states, in fact, that if \(f\) is a \(C^1\) function with compact support in \(\mathbb{R}^n\), then \[ {1\over n} \int_{S^{n-1}} \|\nabla_u f\|^{-n}_1du\leq c_n\| f\|^{-n}_{{n\over n-1}}, \] where \(\nabla_u f\) is the partial derivative of \(f\) in direction \(u\), \(du\) is the standard surface measure on the unit sphere, and the constant \(c_n= ({\omega_n\over 2\omega_{n-1}})^n\) is best. In the latter expression \(\omega_n\) denotes the volume enclosed by the unit sphere \(S^{n-1}\) in \(\mathbb{R}^n\), and \(\| f\|_p\) is the usual \(L_p\) norm of \(f\) in \(\mathbb{R}^n\). This inequality is invariant under the action of \(\text{GL}(n)\) and, as a consequence, does not depend on the Euclidean norm of \(\mathbb{R}^n\). The best constant is attained at the characteristic functions of ellipsoids. A generalization of the Gagliardo-Nirenberg inequality is also proven in the paper.
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 52A39
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 26D10
zbMATH DE Number: 1782621
zbMATH Keywords: Sobolev inequality
zbMATH Keywords: convex bodies
zbMATH Keywords: mixed volume
zbMATH Keywords: surface area element
Wikidata QID: Q115193531
reviewed by: Salvador D. Gigena
MaRDI profile type: Publication
OpenAlex ID: W1547839113
DOI: 10.4310/JDG/1214425451
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description: scientific article; zbMATH DE number 1782621