Previous |  Up |  Next

Article

Keywords:
semiconvexity; rank 1 convexity; polyconvexity; convexity; rotational invariance
Summary:
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 $\tilde{f}.$
References:
[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. MR 1145109
[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 0906209
[3] G. Aubert: Necessary and sufficient conditions for isotropic rank-one convex functions in dimension $2$. J. Elasticity 39 (1995), 31–46. MR 1343149 | Zbl 0828.73015
[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 0573804
[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. MR 0856308
[6] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403. MR 0475169 | Zbl 0368.73040
[7] B. Dacorogna: Direct Methods in the Calculus of Variations. Springer, Berlin, 1989. MR 0990890 | Zbl 0703.49001
[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 1253387
[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 0970519
[10] B. Dacorogna and P. Marcellini: Implicit Partial Differential Equations. Birkhäuser, Basel, 1999. MR 1702252
[11] C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations. Springer, New York, 1966. MR 0202511 | Zbl 0142.38701
[12] P. Rosakis: Characterization of convex isotropic functions. J. Elasticity 49 (1998), 257–267. MR 1633494 | Zbl 0906.73018
[13] M. Šilhavý: The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin, 1997. MR 1423807
[14] M.  Šilhavý: On isotropic rank 1 convex functions. Proc. Roy. Soc. Edinburgh 129A (1999), 1081–1105. MR 1719253
[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 1727470
[16] M. Šilhavý: Rotationally invariant rank 1 convex functions. Appl. Math. Optim. 44 (2001), 1–15. MR 1833365 | Zbl 1032.26007
[17] M. Šilhavý: Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. Royal Soc. Edinburgh 132A (2002), 419–435. MR 1899830 | Zbl 1035.49006
[18] M. Šilhavý: Rank 1 Convex hulls of isotropic functions in dimension 2 by 2. Math. Bohem. 126 (2001), 521–529. MR 1844288 | Zbl 1070.49008
[19] M. Šilhavý: An $O(n)$ invariant rank 1 convex function that is not polyconvex. Theor. Appl. Mech. 28–29 (2002), 325–336. MR 2025155 | Zbl 1055.26006
Partner of
EuDML logo