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Some examples of rank one convex functions in dimension two

Published online by Cambridge University Press:  14 November 2011

B. Dacorogna ,
J. Douchet ,
W. Gangbo  and
J. Rappaz
B. Dacorogna
Affiliation:
Département de Mathématiques, E.P.F.L., 1015 Lausanne, Switzerland
J. Douchet
Affiliation:
Département de Mathématiques, E.P.F.L., 1015 Lausanne, Switzerland
W. Gangbo
Affiliation:
Département de Mathématiques, E.P.F.L., 1015 Lausanne, Switzerland
J. Rappaz
Affiliation:
Département de Mathématiques, E.P.F.L., 1015 Lausanne, Switzerland
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Synopsis

We study the rank one convexity of some functions f(ξ) where ξ is a 2 × 2 matrix. Examples such as |ξ|2α + h(detξ) and | ξ |2α (| ξ |2 − γdet ξ) are investigated. Numerical computations are done on the example of Dacorogna and Marcellini, indicating that this function is quasiconvex.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 1-2 , 1990 , pp. 135 - 150
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Albert, A. A.. A quadratic form problem in the calculus of variations. Bull. Amer. Math. Soc. 44 (1938), 250–253.CrossRefGoogle Scholar
2Aubert, G.. A counterexample of a rank one convex function which is not polyconvex in the case N = 2. Proc. Roy. Soc. Edin., 106 (1987), 237–240.CrossRefGoogle Scholar
3Aubert, G. and Tahraoui, R.. Sur la faible fermeture de certains ensembles de contraintes en é;lasticité non linéaire plan. C. R. Acad. Sci. Paris 290 (1980), 537–540.Google Scholar
4Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 64 (1977), 337–403.Google Scholar
5Ball, J. M.. Does rank one convexity imply quasiconvexity. In Metastability and incompletely posed problems, eds Antman, S. et al. , pp. 17–32 (Berlin: Springer, 1987).CrossRefGoogle Scholar
6Ball, J. M. and Murat, F.. W 1,p quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
7Dacorogna, B.. Direct methods in the calculus of variations (Berlin: Springer, 1989).CrossRefGoogle Scholar
8Dacorogna, B. and Marcellini, P.. A counterexample in the vectorial calculus of variations. In Material Instabilities in Continuum Mechanics, Proceedings, ed. Ball, J. M., pp. 77–83 (Oxford: Oxford Publications 1988).Google Scholar
9Hestenes, M. R. and MacShane, E. J.. A theorem on quadratic forms and its applications in the calculus of variations. Trans. Amer. Math. Soc. 47 (1949), 501–512.CrossRefGoogle Scholar
10Knowles, J. K. and Sternberg, E.. On the ellipticity of the equations of nonlinear elastostatics for a special material. J. Elasticity 5 (1975), 341–361.CrossRefGoogle Scholar
11Kohn, R. and Strang, G.. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math. 39 (1986), 113–137.CrossRefGoogle Scholar
12MacShane, E. J.. The condition of Legendre for double integral problems of the calculus of variations. Bull. Amer. Math. Soc. 45 (1939), Abstract 209, 369.Google Scholar
13Marcellini, P.. Quasiconvex quadratic forms in two dimensions. Appl. Math. Optim. 11 (1984), 183–189.CrossRefGoogle Scholar
14Morrey, C. B.. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
15Morrey, C. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
16Ogden, R. W.. Large deformation isotropic elasticity, on the correlation of theory and experiment for compressible rubberlike solids. Proc. Roy. Soc. London Ser. A 326 (1972), 565–584.Google Scholar
17Reid, W. R.. A theorem on quadratic forms. Bull. Amer. Math. Soc. 44 (1938), 437–440.CrossRefGoogle Scholar
18Serre, D.. Formes quadratiques et calcul des variations J. Math. Pures Appl. 62 (1983), 117–196.Google Scholar
19Terpstra, F. J.. Die Darstellung biquatrischer Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung. Math. Ann. 116 (1938), 166–180.CrossRefGoogle Scholar
20Hove, L. Van. Sur l'extension de la condition de Legendre du calcul des variations aux intégrales multiples à plusieurs fonctions inconnues. Proc. Koninkl. Ned. Akad. 50 (1947), 18–23.Google Scholar