On semiconvexity properties of rotationally invariant functions in two dimensions

Šilhavý, M.

Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 559-571

Abstract (VO)

Let $f$ be a function defined on the set ${\mathbf M}^{2\times 2}$ of all $2$ by $2$ matrices that is invariant with respect to left and right multiplications of its argument by proper orthogonal matrices. The function $f$ can be represented as a function $\tilde{f}$ of the signed singular values of its matrix argument. The paper expresses the ordinary convexity, polyconvexity, and rank 1 convexity of $f$ in terms of its representation $\tilde{f}.$

Détail

Classification : 26B25, 49J10, 49J45, 74B20, 74G65
Keywords: semiconvexity; rank 1 convexity; polyconvexity; convexity; rotational invariance

Bibliographie

[1] J. J. Alibert and B. Dacorogna: An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal. 117 (1992), 155–166. | DOI | MR

[2] G. Aubert: On a counterexample of a rank 1 convex function which is not polyconvex in the case $N=2$. Proc. Roy. Soc. Edinburgh 106A (1987), 237–240. | MR

[3] G. Aubert: Necessary and sufficient conditions for isotropic rank-one convex functions in dimension $2$. J. Elasticity 39 (1995), 31–46. | DOI | MR | Zbl

[4] G. Aubert and R. Tahraoui: Sur la faible fermeture de certains ensembles de contrainte en élasticité nonlinéaire plane. C. R. Acad. Sci. Paris 290 (1980), 537–540. | MR

[5] G. Aubert and R. Tahraoui: Sur la faible fermeture de certains ensembles de contrainte en élasticite nonlinéaire plane. Arch. Rational Mech. Anal. 97 (1987), 33–58. | DOI | MR

[6] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403. | MR | Zbl

[7] B. Dacorogna: Direct Methods in the Calculus of Variations. Springer, Berlin, 1989. | MR | Zbl

[8] B. Dacorogna and H. Koshigoe: On the different notions of convexity for rotationally invariant functions. Ann. Fac. Sci. Toulouse II (1993), 163–184. | MR

[9] B. Dacorogna and P. Marcellini: A counterexample in the vectorial calculus of variations. In: Material Instabilities in Continuum Mechanics, J. M. Ball (ed.), Clarendon Press, Oxford, 1985/1986, pp. 77–83. | MR

[10] B. Dacorogna and P. Marcellini: Implicit Partial Differential Equations. Birkhäuser, Basel, 1999. | MR

[11] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations. Springer, New York, 1966. | MR | Zbl

[12] P. Rosakis: Characterization of convex isotropic functions. J. Elasticity 49 (1998), 257–267. | MR | Zbl

[13] M. Šilhavý: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, 1997. | MR

[14] M.  Šilhavý: On isotropic rank 1 convex functions. Proc. Roy. Soc. Edinburgh 129A (1999), 1081–1105. | MR

[15] M. Šilhavý: Convexity conditions for rotationally invariant functions in two dimensions. In: Applied Nonlinear Analysis, A. Sequeira et al. (ed.), Kluwer Academic, New York, 1999, pp. 513–530. | MR

[16] M. Šilhavý: Rotationally invariant rank 1 convex functions. Appl. Math. Optim. 44 (2001), 1–15. | DOI | MR

[17] M. Šilhavý: Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. Royal Soc. Edinburgh 132A (2002), 419–435. | MR

[18] M. Šilhavý: Rank 1 Convex hulls of isotropic functions in dimension 2 by 2. Math. Bohem. 126 (2001), 521–529. | MR

[19] M. Šilhavý: An $O(n)$ invariant rank 1 convex function that is not polyconvex. Theor. Appl. Mech. 28–29 (2002), 325–336. | MR | Zbl