## 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
Full Text:

### 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 · doi:10.1007/BF00279992 [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 · doi:10.1098/rsta.1982.0095 [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 · doi:10.1112/jlms/s2-26.2.290 [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 · doi:10.2307/1970204 [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 · doi:10.1007/BF00248202 [15] R. Hill, Constitutive inequalities for an isotropic elastic solids under finite strain , Proc. Roy. Soc. London A314 (1970), 457-472. · Zbl 0201.26601 · doi:10.1098/rspa.1970.0018 [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 · doi:10.1007/BF00279991 [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 · doi:10.1112/plms/s3-9.3.432 [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 · doi:10.2307/1989708 [23] 2 H. Whitney, Differentiable even functions , Duke Math. J. 10 (1943), 159-160. · Zbl 0063.08235 · doi:10.1215/S0012-7094-43-01015-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.