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A polyconvexity condition in dimension two

Published online by Cambridge University Press:  14 November 2011

H. Hartwig
H. Hartwig
Affiliation:
Universität Leipzig, FB Mathematik/Informatik, Augustusplatz 10/11, D-04109 Leipzig, Germany
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Extract

We state a necessary and sufficient polyconvexity condition in R2X2 for functions of class C1. This condition is applied to f(X) = |X|2(|X|2 −2 det X) for obtaining a convex representation in R2X2 x R.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 125 , Issue 5 , 1995 , pp. 901 - 910
Copyright
Copyright © Royal Society of Edinburgh 1995

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References

1Alibert, J. J. and Dacorogna, B.. An example of a quasiconvex function that is not polyconvex in two dimensions. Arch. Rational Mech. Anal. 117 (1992), 155–66.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 64 (1977), 337–403.Google Scholar
3Dacorogna, B.. Direct Methods in the Calculus of Variations (Berlin: Springer, 1989).CrossRefGoogle Scholar
4Dacorogna, B., Douchet, J., Gangbo, W. and Rappaz, J.. Some examples of rank-one convex functions in dimension two. Proc. Roy. Soc. Edinburgh Sect. A 114 (1990), 135–50.CrossRefGoogle Scholar
5Dacorogna, B. and Koshigoe, H.. On the different notions of convexity for rotationally invariant functions. Ann. Fac. Sci. Toulouse II (1993), 163–84.CrossRefGoogle Scholar
6Dacorogna, B. and Marcellini, P.. A counterexample in the vectorial calculus of variations. In Material Instabilities in Continuum Mechanics. Proceedings, ed. Ball, J. M., 77–83 (Oxford: Oxford Publications, 1988).Google Scholar
7Hartwig, H.. Quasiconvexity and related properties in the calculus of variations. Proceedings of the IVth International Workshop on Generalized Convexity, Pécs, 1992 (Berlin: Springer, 1994).Google Scholar
8Iwaniec, T. and Lutoborski, L.. Integral estimates for null Lagrangians. Arch. Rational Mech. Anal. 125 (1993), 25–79.CrossRefGoogle Scholar
9Kohn, R. and Strang, G.. Optimal design and relaxation of variational problems. Cotnm. Pure Appl. Math. 39 (1986), 113–37; 139–82; 353–77.CrossRefGoogle Scholar
10Morrey, C. B.. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
11Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
12Šverák, V.. Quasiconvex functions with subquadratic growth. Proc. Roy. Soc. London Ser. A 433 (1991), 723–5.Google Scholar
13Šverák, V.. Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120(1992), 185–9.CrossRefGoogle Scholar