Differentiability properties of symmetric and isotropic functions. (English) Zbl 0566.73001

Ball, J. M.

Duke Math. J. 51, 699-728 (1984).

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.

Reviewer: Vadim Komkov (Florida)

MSC:

58C25	Differentiable maps on manifolds
26C05	Real polynomials: analytic properties, etc.
74E15	Crystalline structure
74B20	Nonlinear elasticity

Keywords:

symmetric functions; relations between differentiability properties; Dieudonné’s extension theorem; Whitney’s regularity theorem; stored energy function; frame indifference postulate; isotropy

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