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On SO(n)-Invariant Rank 1 Convex Functions

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Abstract

Let f be a function defined on the set M n×n of all real square matrices of order n. If f is SO(n)-invariant, it has a representation f on R n through the signed singular values of the matrix argument A∈M n×n. A necessary and sufficient condition for the rank 1 convexity of f in terms of f is given.

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References

  1. G. Aubert, Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2. J. Elasticity 39 (1995) 31–46.

    MATH  MathSciNet  Google Scholar 

  2. G. Aubert and R. Tahraoui, Sur la faible fermeture de certains ensembles de contraintes en élasticité non linéaire plane. Arch. Rational Mech. Anal. 97 (1987) 33–58.

    MATH  MathSciNet  ADS  Google Scholar 

  3. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977) 337–403.

    MATH  ADS  Google Scholar 

  4. J.M. Ball, Differentiability properties of symmetric and isotropic functions. Duke Math. J. 51 (1984) 699–728.

    Article  MATH  MathSciNet  Google Scholar 

  5. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal. 100 (1987) 13–52.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two-well problem. Philos. Trans. Roy. Soc. London 338 (1992) 389–450.

    MATH  ADS  Google Scholar 

  7. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Rational Mech. Anal. 103 (1988) 237–277.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. B. Dacorogna, Direct methods in the calculus of variations. Springer, Berlin (1989).

    MATH  Google Scholar 

  9. B. Dacorogna, Necessary and sufficient conditions for strong ellipticity of isotropic functions in any dimension. Discrete Contin. Dyn. Syst. B2 (2001) 257–263.

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. B. Dacorogna and P. Marcellini, Implicit Partial Differential Equations. Birkhäuser, Basel (1999).

    MATH  Google Scholar 

  12. J.K. Knowles and E. Sternberg, On the failure of ellipticity of the equations for finite elastostatic plane strain. Arch. Rational Mech. Anal. 63 (1977) 321–326.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems, I, II, III. Comm. Pure Appl. Math. 39 (1986) 113–137, 139–182, 353–377.

    MATH  MathSciNet  Google Scholar 

  14. C.B. Morrey Jr, Multiple Integrals in the Calculus of Variations. Springer, New York (1966).

    MATH  Google Scholar 

  15. S. Müller, Variational models for microstructure and phase transitions. In: Calculus of Variations and Geometric Evolution Problems (Cetraro, 1996), Lecture Notes in Math. 1713. Springer, Berlin (1999) pp. 85–210.

    Google Scholar 

  16. P. Pedregal, Parametrized Measures and Variational Principles. Birkhäuser, Basel (1997).

    MATH  Google Scholar 

  17. P. Rosakis, Characterization of convex isotropic functions. J. Elasticity 49 (1997) 257–267.

    Article  MathSciNet  Google Scholar 

  18. T. Roubíček, Relaxation in Optimization Theory and Variational Calculus. W. de Gruyter, Berlin (1997).

    MATH  Google Scholar 

  19. M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media. Springer, Berlin (1997).

    MATH  Google Scholar 

  20. M. Šilhavý, Convexity conditions for rotationally invariant functions in two dimensions. In: A. Sequeira et al. (eds), Applied Nonlinear Analysis. Kluwer Academic Publishers, New York (1999) pp. 513–530.

    Google Scholar 

  21. M. Šilhavý, On isotropic rank 1 convex functions. Proc. Roy. Soc. Edinburgh A 129 (1999) 1081–1105.

    MATH  Google Scholar 

  22. M. Šilhavý, Rotationally invariant rank 1 convex functions. Appl.Math. Optim. 44 (2001) 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Šilhavý, Rank 1 convex hulls of isotropic functions in dimension 2 by 2. Math. Bohem. 126 (2001) 521–529.

    MathSciNet  MATH  Google Scholar 

  24. M. Šilhavý, Monotonicity of rotationally invariant convex and rank 1 convex functions. Proc. Roy. Soc. Edinburgh A 132 (2001) 419–435.

    Google Scholar 

  25. M. Šilhavý, On the semiconvexity properties of rotationally invariant functions in two dimensions. (2001) To be published.

  26. M. Šilhavý, Rank 1 convex hulls of rotationally invariant functions. In: Ch. Miehe (ed.), Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Kluwer Academic Publishers, New York (2003) pp. 87–98. In press.

    Google Scholar 

  27. H.C. Simpson and S. Spector, On copositive matrices and strong ellipticity for isotropic elastic materials. Arch. Rational Mech. Anal. 84 (1983) 55–68.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. C. Truesdell and W. Noll, The non-linear field theories of mechanics. In: S. Fluegge (ed.), Handbuch der Physik III/3. Springer, Berlin (1965).

    Google Scholar 

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  1. Mathematical Institute of the AV ČR, Žitná 25, 115 67, Prague 1, Czech Republic

    M. Šilhavý

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Šilhavý, M. On SO(n)-Invariant Rank 1 Convex Functions. Journal of Elasticity 71, 235–246 (2003). https://doi.org/10.1023/B:ELAS.0000005544.24267.8d

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  • DOI: https://doi.org/10.1023/B:ELAS.0000005544.24267.8d

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