Abstract

We give direct estimates for the quasiconvex polytopes Q(K)Q(K) generated by a finite set K⊂MN×nK\subset M^{N\times n}. More precisely, we bound the quasiconvex envelope Qdist⁡(⋅,K)Q\dist(\cdot,K) near a convex exposed face of C(X)C(X) which does not have rank-one connections. Our estimates depend on the weak-(1,1) bounds for certain singular integral operators and the geometric features of the convex polytope C(K)C(K). We show by an example that our estimate is `local' and independent of the `size' of KK, hence it is a better estimate than the polyconvex hull P(K)P(K) which is `size' dependent.

Author information

Contact details are reproduced from the original publication and may be historical.

Kewei Zhang

Dept. of Mathematics, University of Sussex, Falmer, Brighton BN1 9RF, Great Britain

k.zhang@sussex.ac.uk

Keywords

  • Direct estimates
  • quasiconvex polytopes
  • quasiconvex envelope
  • singular integral operators
  • polyconvex hull

References

  1. J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity, Arch. Ration. Mech. Anal. 63 (1977) 337–403.https://doi.org/10.1007/BF00279992Google Scholar
  2. J. M. Ball, R. D. James: Fine phase mixtures as minimizers of energy, Arch. Ration. Mech. Anal. 100 (1987) 13–52.https://doi.org/10.1007/BF00281246Google Scholar
  3. J. M. Ball, R. D. James: Proposed experimental tests of a theory of fine microstructure and the two-well problem, Philos. Trans. R. Soc. Lond., Ser. A 338 (1992) 389–450.https://doi.org/10.1098/rsta.1992.0013Google Scholar
  4. K. Bhattacharya, N. B. Firoozye, R. D. James, R. V. Kohn: Restrictions on microstructure, Proc. R. Soc. Edinb., Sect. A 124 (1994) 843–878.https://doi.org/10.1017/S0308210500022381Google Scholar
  5. A. Brøndsted: An Introduction to Convex Polytopes, Springer, New York (1983).https://doi.org/10.1007/978-1-4612-1148-8Google Scholar
  6. B. Dacorogna: Direct Methods in the Calculus of Variations, Springer, Berlin (1989).https://doi.org/10.1007/978-3-642-51440-1Google Scholar
  7. G. Dolzmann, B. Kirchheim, J. Kristensen: Conditions for equility of hulls in the calculus of variations, Arch. Ration. Mech. Anal. 154 (2000) 93–100.https://doi.org/10.1007/s002050000098Google Scholar
  8. I. Ekeland, R. Temam: Convex Analysis and Variational Problems, North-Holland, Amsterdam (1976).Google Scholar
  9. Z. Iqbal: Variational Methods in Solid Mechanics, PhD Thesis, University of Oxford (1999).Google Scholar
  10. C. B. Morrey, Jr.: Multiple Integrals in the Calculus of Variations, Springer, Berlin (1966).https://doi.org/10.1007/978-3-540-69952-1Google Scholar
  11. S. Müller: On quasiconvex functions which are homogeneous of degree one, Indiana Math. J. 41 (1992) 295–300.https://doi.org/10.1512/iumj.1992.41.41017Google Scholar
  12. 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) 85–210.https://doi.org/10.1007/BFb0092670Google Scholar
  13. S. Müller, V. Šverák: Attainment results for the two-well problem by convex integration, in: Geometric Analysis and the Calculus of Variations, Internat. Press, Cambridge (1996) 239–251.Google Scholar
  14. R. T. Rockafellar: Convex Analysis, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400873173Google Scholar
  15. E. M. Stein: Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton (1970).https://doi.org/10.1515/9781400883882Google Scholar
  16. E. M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton (1993).https://doi.org/10.1515/9781400883929Google Scholar
  17. V. Šverák: On the problem of two wells, in: Microstructure and Phase Transition, D. Kinderlehrer et al. (eds.), IMA Vol. Math. Appl. 54, Springer, New York (1993) 183–189.https://doi.org/10.1007/978-1-4613-8360-4_11Google Scholar
  18. V. Šverák: On regularity of the Monge-Ampère equation without convexity assumptions, preprint (1990).Google Scholar
  19. V. Šverák: New examples of quasiconvex functions, Arch. Ration. Mech. Anal. 119 (1992) 293–330.https://doi.org/10.1007/BF01837111Google Scholar
  20. L. Tartar: Some remarks on separately convex functions, in: Microstructure and Phase Transition, D. Kinderlehrer et al. (eds.), IMA Vol. Math. Appl. 54, Springer, New York (1993) 191–204.https://doi.org/10.1007/978-1-4613-8360-4_12Google Scholar
  21. K.-W. Zhang: Quasiconvex functions, SO(n) and two elastic wells, Ann. Inst. H. Poincaré, Anal. Non Linéaire 14 (1997) 759–785.https://doi.org/10.1016/s0294-1449(97)80132-1Google Scholar
  22. K.-W. Zhang: On various semiconvex hulls in the calculus of variations, Calc. Var. Partial Differ. Equ. 6 (1998) 143–160.https://doi.org/10.1007/s005260050086Google Scholar
  23. K.-W. Zhang: On some quasiconvex functions with linear growth, J. Convex Analysis 5 (1998) 133–146.https://doi.org/10.68381/jca05-10Google Scholar

Suggested citation

K. Zhang. “Estimates of Quasiconvex Polytopes in the Calculus of Variations.” Journal of Convex Analysis 13 (2006), No. 1, 37–50. https://doi.org/10.68381/jca13003

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Published by Heldermann Verlag, 2006. Rights now held by Banach Press LLC.