## 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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### References:

 [1] V. I. Arnold, Geometrical methods in the theory of ordinary differential equations , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 250, Springer-Verlag, New York, 1983. · Zbl 0507.34003 [2] A. Aubert and R. Tahraoui, Conditions necessaires de faible fermeture et de $$1$$-rang convexité en dimension $$3$$ , preprint, 1982. · Zbl 0647.73017 [3] 1 J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity , Arch. Rat. Mech. Anal. 63 (1976/77), no. 4, 337-403. · Zbl 0368.73040 [4] 2 J. M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity , Philos. Trans. Roy. Soc. London Ser. A 306 (1982), no. 1496, 557-611. JSTOR: · Zbl 0513.73020 [5] G. Barbançon, A propos du théorème de Newton pour les fonctions de classe $$C^m$$ et d’une generalization de la notion de multiplicateur rugueux , Ann. Fac. Sci. Phnom Penh (1969), Théorème de Newton pour les fonctions de classe $$C^r$$ Ann. Scient. Ec. Norm. Sup. 4ème Série 5 (1972), 435-458. [6] G. Barbançon and M. Rais, Sur le théorème de Hilbert différentiable pour les groupes linéaires finis (d’après E. Noether) , preprint, 1982. [7] J. Dieudonné, Foundations of Modern Analysis , Pure and Applied Mathematics, Vol. X, Academic Press, New York, 1960. · Zbl 0100.04201 [8] H. Federer, Geometric measure theory , Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. · Zbl 0176.00801 [9] L. E. Fraenkel, On the embedding of $$C^1(\bar \Omega )$$ in $$C^0,\alpha (\bar \Omega )$$ , J. London Math. Soc. (2) 26 (1982), no. 2, 290-298. · Zbl 0503.46017 [10] F. R. Gantmacher, The theory of matrices. Vols. 1, 2 , Translated by K. A. Hirsch, Chelsea Publishing Co., New York, 1959. · Zbl 0085.01001 [11] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order , Springer-Verlag, Berlin, 1977. · Zbl 0361.35003 [12] G. Glaeser, Fonctions composées différentiables , Ann. of Math. (2) 77 (1963), 193-209. · Zbl 0106.31302 [13] M. E. Gurtin, An introduction to continuum mechanics , Mathematics in Science and Engineering, vol. 158, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1981. · Zbl 0559.73001 [14] M. Hayes, Static implications of the strong-ellipticity condition , Arch. Rat. Mech. Anal. 33 (1969), 181-191. · Zbl 0201.26502 [15] R. Hill, Constitutive inequalities for an isotropic elastic solids under finite strain , Proc. Roy. Soc. London A314 (1970), 457-472. · Zbl 0201.26601 [16] T. Kato, Perturbation theory for linear operators , Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. · Zbl 0148.12601 [17] J. K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain , Arch. Rational Mech. Anal. 63 (1976), no. 4, 321-336 (1977). · Zbl 0351.73061 [18] F. Rellich, Perturbation Theory of Eigenvalue Problems , Assisted by J. Berkowitz. With a preface by Jacob T. Schwartz, Gordon and Breach Science Publishers, New York, 1969. · Zbl 0181.42002 [19] J. Serrin, The derivation of stress-deformation relations for a Stokesian fluid. , J. Math. Mech. 8 (1959), 459-469. · Zbl 0089.18601 [20] C. Truesdell and W. Noll, The nonlinear field theories of mechanics , Handbuch der Physik ed. S. Flügge, vol. III/3, Springer, Berlin, 1965. · Zbl 0779.73004 [21] H. D. Ursell, Inequalities between sums of powers , Proc. London Math. Soc. (3) 9 (1959), 432-450. · Zbl 0090.01504 [22] 1 H. Whitney, Analytic extensions of differentiable functions defined in closed sets , Trans. Amer. Math. Soc. 36 (1934), no. 1, 63-89, Functions differentiable on the boundaries of regions, Ann. of Math. 55 (1934b). JSTOR: · Zbl 0008.24902 [23] 2 H. Whitney, Differentiable even functions , Duke Math. J. 10 (1943), 159-160. · Zbl 0063.08235
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