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

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.

MSC:

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