On the structure of quasiconvex hulls

Kewei Zhang

Annales de l'I.H.P. Analyse non linéaire (1998)

  • Volume: 15, Issue: 6, page 663-686
  • ISSN: 0294-1449

How to cite


Zhang, Kewei. "On the structure of quasiconvex hulls." Annales de l'I.H.P. Analyse non linéaire 15.6 (1998): 663-686. <http://eudml.org/doc/78452>.

author = {Zhang, Kewei},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {quasiconvexity; convexity; extreme points; convex hull; gradient homogeneous Young measures; Dirac maps; quasiconvex hull},
language = {eng},
number = {6},
pages = {663-686},
publisher = {Gauthier-Villars},
title = {On the structure of quasiconvex hulls},
url = {http://eudml.org/doc/78452},
volume = {15},
year = {1998},

AU - Zhang, Kewei
TI - On the structure of quasiconvex hulls
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 1998
PB - Gauthier-Villars
VL - 15
IS - 6
SP - 663
EP - 686
LA - eng
KW - quasiconvexity; convexity; extreme points; convex hull; gradient homogeneous Young measures; Dirac maps; quasiconvex hull
UR - http://eudml.org/doc/78452
ER -


  1. [A] E.M. Alfsen, Compact Convex Sets and Boundary Integrals, Springer-Verlag, 1971. Zbl0209.42601MR445271
  2. [AF] E. Acerbi and N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., Vol. 86, 1984, pp. 125-145. Zbl0565.49010MR751305
  3. [BL] H. Berliocchi, J.M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations. Bull. Soc. Math. France, Vol. 101, 1973, pp. 129-184. Zbl0282.49041MR344980
  4. [B11] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., Vol. 63, 1977, pp. 337-403. Zbl0368.73040MR475169
  5. [B12] J.M. Ball, A version of the fundamental theorem of Young measures, in Partial Differential Equations and Continuum Models of Phase Transitions, (edited by M. RASCLE, D. SERRE and M. SLEMROD), 1989, pp. 207-215, Springer-Verlag. Zbl0991.49500MR1036070
  6. [B13] J.M. Ball, Sets of gradients with no rank-one connections. J. Math. Pures et Appl., Vol. 69, 1990, pp. 241-259. Zbl0644.49011MR1070479
  7. [BFJK] K. Bhattacharya, N.B. Firoozye, R.D. James, R.V. Kohn, Restrictions on Microstructures. Proc. Royal Soc. Edinburgh, A, Vol. 124, 1994, pp. 843-878. Zbl0808.73063MR1303758
  8. [BJ1] J.M. Ball, R.D. James, Fine phase mixtures as minimizers of energy. Arch. Rational Mech. Anal., Vol. 100, 1987, pp. 13-52. Zbl0629.49020MR906132
  9. [BJ2] J.M. Ball, R.D. James, Proposed experimental tests of a theory of fine microstructures and the two-well problem. Phil. Roval Soc. Lon., Vol. 338A, 1992, pp. 389-450. Zbl0758.73009
  10. [BZ] J.M. Ball and K.-W. Zhang, Lower semicontinuity and multiple integrals and the biting lemma . Proc. Royal Soc. Edinburgh, Vol. 114A, 1990, pp. 367-379. Zbl0716.49011MR1055554
  11. [CK] M. Chipot, D. Kinderlehrer, Equilibrium configurations of crystalsArch. Rational Mech. Anal., Vol. 103, 1988, pp. 237-277. Zbl0673.73012MR955934
  12. [D] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, 1989. Zbl0703.49001MR990890
  13. [ET] I. Ekeland, R. Temam, Convex Aanlysis and Variational Problems, North-Holland, 1976. Zbl0322.90046MR463994
  14. [K] D. Kinderlehrer, Remarks about equilibrium configurations of crystals, in Material Instabilities in Continuum Mechanics, J. M. BALL ed., Oxford University Press, 1988, pp. 977-83. Zbl0850.73037MR970527
  15. [KP] D. Kinderlehrer, P. Pedregal, Characterizations of Young measures generated by gradients. Arch. Rational Mech. Anal, Vol. 115, 1991, pp. 329-365. Zbl0754.49020MR1120852
  16. [Ma] J.P. Matos, Young measures and the absence of fine microstructures in a class of phase transitions. European J. Appl. Math, Vol. 3, 1992, pp. 31-54. Zbl0751.73003MR1156593
  17. [Mo] C.B. Jr Morrey, Multiple integrals in the calculus of variations, Springer, 1966. Zbl0142.38701MR202511
  18. [MS] S. Müller, V. Šverák, Attainment results for the two-well problem by convex integration, preprint, 1993. 
  19. [Re] Yu.G. Reshetnak, Liouville's theorem on conformal mappings under minimal regularity assumptions. Siberian Math. J., Vol. 8, 1967, pp. 631-653. Zbl0167.36102
  20. [Ro] R.T. Rockafellar, Convex Analysis, Princeton University Press, 1970. Zbl0193.18401MR274683
  21. [Ru] W. Rudin, Functional Analysis, McGraw-Hill, 1973. Zbl0253.46001MR365062
  22. [Sv1] V. Šverák, On the problem of two wells, preprint. MR1320537
  23. [Sv2] V. Šverák, On Tartar's conjecture. Ann. Inst. H. Poincaré, Vol. 10, 1993, pp. 405-412. Zbl0820.35022MR1246459
  24. [Sv3] V. Šverák, Rank one convexity does not imply quasiconvexity. Proc. Royal Soc. Edin., Vol. 120A, 1992, pp. 185-189. Zbl0777.49015MR1149994
  25. [T] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, R. J. Knops ed Pitman, 1979. Zbl0437.35004MR584398
  26. [Y] B.-S. Yan, Remarks on the set of quasi-conformal matrices in higher dimensions, Preprint, 1994. 
  27. [Z1] K.-W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm. Sup. Pisa Serie IV , Vol. XIX, 1992, pp. 313-326. Zbl0778.49015MR1205403
  28. [Z2] K.-W. Zhang, On non-negative quasiconvex functions with unbounded zero sets, Proc. Royal Soc. Edin., Vol. 127A, 1997, pp. 411-422. Zbl0883.49013MR1447961
  29. [Z3] K.-W. Zhang, On some quasiconvex functions with linear growth, to appear in J. Convex Anal. Zbl0915.49008MR1649465

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