Differentiability properties of symmetric and isotropic functions
type: Item
instance of: scholarly article
title: Differentiability properties of symmetric and isotropic functions
zbMATH Open document ID: 0566.73001
DOI: 10.1215/S0012-7094-84-05134-2
published in: Duke Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Let \(\mathbb{G}\) be a group of transformations \(\mathbb{R}^n\to \mathbb{R}^n\). A function \(f\) is invariant under \(\mathbb{G}\) if for any \(x\in \mathbb{R}^n\) it is true that \(f(Tx)=f(x)\), \(T\in \mathbb{G}\). Several characterizations of invariance have the form: let \(x\to y\) be a transformation of variables, then there exists a function \(F\) such that \(F(y(x))=f(x)\), \(\forall x\in \mathbb{R}^n\). For example, the function \(f\) is invariant under the transformation \(x\to ix\) if and only if there exists \(F\) such that \(f(x)=F(x^4)\). The main purpose of this paper is to establish relations between differentiability properties of \(f\) and \(F\). In general one expects to loose some differentiability by going from \(f\) to \(F\). The author uses a version of Dieudonné's extension theorem and Whitney's regularity theorem to prove several relations between the differentiability of\(f\) and \(F\). The final section of the paper contains an application to nonlinear elasticity. The author proves differentiability of the ''stored energy function'' if one assumes the frame indifference postulate and isotropy.
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 26C05
Mathematics Subject Classification ID: 74E15
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 3904039
zbMATH Keywords: symmetric functions
zbMATH Keywords: relations between differentiability properties
zbMATH Keywords: Dieudonné's extension theorem
zbMATH Keywords: Whitney's regularity theorem
zbMATH Keywords: stored energy function
zbMATH Keywords: frame indifference postulate
zbMATH Keywords: isotropy
author: J. M. Ball
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3904039