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Volume 12, Number 1, January 2006
Page(s) 64 - 92
Published online 15 December 2005
  1. J.J. Alibert and B. Dacorogna, An example of a quasiconvex function that is not polyconvex in two dimensions two. Arch. Ration. Mech. Anal. 117 (1992) 155–166. [Google Scholar]
  2. S. Agmon, Maximum theorems for solutions of higher order elliptic equations. Bull. Am. Math. Soc. 66 (1960) 77–80. [CrossRef] [Google Scholar]
  3. S. Agmon, L. Nirenberg and M.H. Protter, A maximum principle for a class of hyperbolic equations and applications to equations of mixed elliptic–hyperbolic type. Commun. Pure Appl. Math. 6 (1953) 455-470. [CrossRef] [Google Scholar]
  4. K. Astala, Analytic aspects of quasiconformality. Doc. Math. J. DMV, Extra Volume ICM, Vol. II (1998) 617–626. [Google Scholar]
  5. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977) 337–403. [Google Scholar]
  6. J.M. Ball, Constitutive inequalities and existence theorems in nonlinear elastostatics. Nonlin. Anal. Mech., Heriot–Watt Symp. Vol. I, R. Knops Ed. Pitman, London (1977) 187–241. [Google Scholar]
  7. J.M. Ball and R.D. James, Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100 (1987) 13–52. [Google Scholar]
  8. J.M. Ball and R.D. James, Proposed experimental tests of a theory of fine microstructure and the two–well problem. Philos. Trans. R. Soc. Lond. 338(A) (1992) 389–450. [Google Scholar]
  9. J.M. Ball, B. Kirchheim and J. Kristensen, Regularity of quasiconvex envelopes. Calc. Var. Partial Differ. Equ. 11 (2000) 333–359. [Google Scholar]
  10. J.M. Ball and F. Murat, Remarks on rank-one convexity and quasiconvexity. Ordinary and Partial Differential Equations, B.D. Sleeman and R.J. Jarvis Eds. Vol. III, Longman, New York. Pitman Res. Notes Math. Ser. 254 (1991) 25–37. [Google Scholar]
  11. J.M. Ball, J.C. Currie and P.J. Olver, Null Lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal. 41 (1981) 135–174. [CrossRef] [MathSciNet] [Google Scholar]
  12. A.V. Bitsadze, A system of nonlinear partial differential equations. Differ. Uravn. 15 (1979) 1267–1270 (in Russian). [Google Scholar]
  13. A. Canfora, Teorema del massimo modulo e teorema di esistenza per il problema di Dirichlet relativo ai sistemi fortemente ellittici. Ric. Mat. 15 (1966) 249–294. [Google Scholar]
  14. E. Casadio-Tarabusi, An algebraic characterization of quasiconvex functions. Ric. Mat. 42 (1993) 11–24. [Google Scholar]
  15. M. Chipot and D. Kinderlehrer, Equilibrium configurations of crystals. Arch. Ration. Mech. Anal. 103 (1988) 237–277. [Google Scholar]
  16. B. Dacorogna, Weak continuity and weak lower semicontinuity for nonlinear functionals. Berlin-Heidelberg-New York, Springer. Lect. Notes Math. 922 (1982). [Google Scholar]
  17. B. Dacorogna, Direct methods in the calculus of variations. Springer, Berlin (1989). [Google Scholar]
  18. B. Dacorogna, J. Douchet, W. Gangbo and J. Rappaz, Some examples of rank–one convex functions in dimension two. Proc. R. Soc. Edinb. 114 (1990) 135–150. [Google Scholar]
  19. B. Dacorogna and J.-P. Haeberly, Some numerical methods for the study of the convexity notions arising in the calculus of variations. M2AN 32 (1998) 153–175. [Google Scholar]
  20. G. Dolzmann, Numerical computation of rank–one convex envelopes. SIAM J. Numer. Anal. 36 (1999) 1621–1635. [CrossRef] [MathSciNet] [Google Scholar]
  21. G. Dolzmann, Variational methods for crystalline microstructure–analysis and computation. Springer-Verlag, Berlin. Lect. Notes Math. 1803 (2003). [Google Scholar]
  22. G. Dolzmann, B. Kirchheim and J. Kristensen, Conditions for equality of hulls in the calculus of variations. Arch. Ration. Mech. Anal. 154 (2000) 93–100. [CrossRef] [MathSciNet] [Google Scholar]
  23. D.G.B. Edelen, The null set of the Euler–Lagrange operator. Arch. Ration. Mech. Anal 11 (1962) 117–121. [CrossRef] [Google Scholar]
  24. H. Federer, Geometric measure theory. Springer-Verlag, New York, Heldelberg (1969). [Google Scholar]
  25. I. Fonseca and S. Müller, A–quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30 (1999) 1355–1390. [CrossRef] [MathSciNet] [Google Scholar]
  26. L.E. Fraenkel, An introduction to maximum principles and symmetry in elliptic problems. Cambridge University Press, Cambridge (2000). [Google Scholar]
  27. M. Giaquinta and E. Giusti, Quasi–minima, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1 (1984) 79–107. [Google Scholar]
  28. D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin–Heidelberg–New York (1977). [Google Scholar]
  29. T. Iwaniec, Nonlinear Cauchy–Riemann operators in Formula . Trans. Am. Math. Soc. 354 (2002) 1961–1995. [Google Scholar]
  30. T. Iwaniec, Integrability theory of the Jacobians. Lipshitz Lectures, preprint Univ. Bonn Sonderforschungsbereich 256 (1995). [Google Scholar]
  31. T. Iwaniec, Nonlinear differential forms. Series in Lectures at the International School in Jyväskylä, published by Math. Inst. Univ. Jyväskylä (1998) 1–207. [Google Scholar]
  32. T. Iwaniec and A. Lutoborski, Integral estimates for null–lagrangians. Arch. Ration. Mech. Anal. 125 (1993) 25–79. [CrossRef] [Google Scholar]
  33. A. Kałamajska, On Formula –convexity conditions in the theory of lower semicontinuous functionals. J. Convex Anal. 10 (2003) 419–436. [Google Scholar]
  34. A. Kałamajska, On new geometric conditions for some weakly lower semicontinuous functionals with applications to the rank-one conjecture of Morrey. Proc. R. Soc. Edinb. A 133 (2003) 1361–1377. [CrossRef] [Google Scholar]
  35. B. Kirchheim, S. Müller and V. Šverák, Studing nonlinear pde by geometry in matrix space, in Geometric Analysis and Nonlinear Differential Equations, H. Karcher and S. Hildebrandt Eds. Springer (2003) 347–395. [Google Scholar]
  36. V. Kohn and G. Strang, Optimal design and relaxation of variational problems I. Commun. Pure Appl. Math. 39 (1986) 113–137. [Google Scholar]
  37. V. Kohn and G. Strang, Optimal design and relaxation of variational problems II. Commun. Pure Appl. Math. 39 (1986) 139–182. [Google Scholar]
  38. J. Kolář, Non–compact lamination convex hulls. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003) 391–403. [Google Scholar]
  39. J. Kristensen, On the non–locality of quasiconvexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 16 (1999) 1–13. [Google Scholar]
  40. M. Kružík, On the composition of quasiconvex functions and the transposition. J. Convex Anal. 6 (1999) 207–213. [Google Scholar]
  41. M. Kružík, Bauer's maximum principle and hulls of sets. Calc. Var. Partial Differ. Equ. 11 (2000) 321–332. [Google Scholar]
  42. S. Lang, Algebra. Addison–Wesley Publishing Company, New York (1965). [Google Scholar]
  43. H. Le Dret and A. Raoult, The quasiconvex envelope of the Saint–Venant–Kirchhoff stored energy function. Proc. R. Soc. Edinb. 125 (1995) 1179–1192. [Google Scholar]
  44. F. Leonetti, Maximum principle for vector–valued minimizers of some integral functionals. Boll. Unione Mat. Ital. 7 (1991) 51–56. [Google Scholar]
  45. P.L. Lions, Jacobians and Hardy spaces. Ric. Mat. Suppl. 40 (1991) 255–260. [Google Scholar]
  46. M. Luskin, On the computation of crystalline microstructure. Acta Numerica 5 (1996) 191–257. [CrossRef] [Google Scholar]
  47. J.J. Manfredi, Weakly monotone functions. J. Geom. Anal. 4 (1994) 393–402. [CrossRef] [MathSciNet] [Google Scholar]
  48. P. Marcellini, Quasiconvex quadratic forms in two dimensions. Appl. Math. Optimization 11 (1984) 183–189. [Google Scholar]
  49. M. Miranda, Maximum principles and minimal surfaces. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) XXV (1997) 667–681. [Google Scholar]
  50. C.B. Morrey, Quasi–convexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2 (1952) 25–53. [Google Scholar]
  51. C.B. Morrey, Multiple integrals in the calculus of variations. Springer-Verlag, Berlin–Heidelberg–New York (1966). [Google Scholar]
  52. F. Murat, Compacité par compensation. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 5 (1978) 489–507. [Google Scholar]
  53. F. Murat, A survey on compensated compactness. Contributions to modern calculus of variations, L. Cesari Ed. Longman, Harlow, Pitman Res. Notes Math. Ser. 148 (1987) 145–183. [Google Scholar]
  54. F. Murat, Compacité par compensation; condition nécessaire et suffisante de continuité faible sous une hypothése de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 8 (1981) 69–102. [Google Scholar]
  55. S. Müller, A surprising higher integrability property of mappings with positive determinant. Bull. Am. Math. Soc. 21 (1989) 245–248. [CrossRef] [Google Scholar]
  56. S. Müller, Variational models for microstructure and phase transitions, Collection: Calculus of variations and geometric evolution problems (Cetraro 1996), Springer, Berlin. Lect. Notes Math. 1713 (1999) 85–210. [CrossRef] [Google Scholar]
  57. S. Müller, Rank–one convexity implies quasiconvexity on diagonal matrices. Int. Math. Res. Not. 20 (1999) 1087–1095. [Google Scholar]
  58. S. Müller, Quasiconvexity is not invariant under transposition, in Proc. R. Soc. Edinb. 130 (2000) 389–395. [Google Scholar]
  59. S. Müller and V. Šverák, Attainment results for the two–well problem by convex integration. Geom. Anal. and the Calc. Variations, J. Jost Ed. International Press (1996) 239–251. [Google Scholar]
  60. S. Müller and V. Šverák, Unexpected solutions of first and second order partial differential equations. Doc. Math. J. DMV, Special volume Proc. ICM, Vol. II (1998) 691–702. [Google Scholar]
  61. S. Müller and V. Šverák, Convex integration for Lipschitz mappings and counterexamples to regularity. Ann. of Math. (2) 157 (2003) 715–742. [Google Scholar]
  62. S. Müller and V. Šverák, Convex integration with constrains and applications to phase transitions and partial differential equations. J. Eur. Math. Soc. (JEMS) 1/4 (1999) 393–422. [Google Scholar]
  63. S. Müller and M.O. Rieger, V. Šverák, Parabolic systems with nowhere smooth solutions, preprint, [Google Scholar]
  64. G.P. Parry, On the planar rank–one convexity condition. Proc. R. Soc. Edinb. A 125 (1995) 247–264. [Google Scholar]
  65. P. Pedregal, Parametrized measures and variational principles. Birkhäuser (1997). [Google Scholar]
  66. P. Pedregal, Weak continuity and weak lower semicontinuity for some compensation operators. Proc. R. Soc. Edinb. A 113 (1989) 267–279. [Google Scholar]
  67. P. Pedregal, Laminates and microstructure. Eur. J. Appl. Math. 4 (1993) 121–149. [Google Scholar]
  68. P. Pedregal, Some remarks on quasiconvexity and rank–one convexity. Proc. R. Soc. Edinb. A 126 (1996) 1055–1065. [Google Scholar]
  69. P. Pedregal and V. Šverák, A note on quasiconvexity and rank–one convexity for Formula Matrices. J. Convex Anal. 5 (1998) 107–117. [Google Scholar]
  70. A.C. Pipkin, Elastic materials with two preferred states. Q. J. Mech. Appl. Math. 44 (1991) 1–15. [Google Scholar]
  71. J. Robbin, R.C. Rogers and B. Temple, On weak continuity and Hodge decomposition. Trans. Am. Math. Soc. 303 (1987) 609–618. [CrossRef] [Google Scholar]
  72. T. Roubíuek, Relaxation in optimization theory and variational calculus. Berlin, W. de Gruyter (1997). [Google Scholar]
  73. J. Sivaloganathan, Implications of rank one convexity. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 5, 2 (1988) 99–118. [Google Scholar]
  74. R. Stefaniuk, Numerical verification of certain property of quasiconvex function. MSC thesis, Warsaw University (2004). [Google Scholar]
  75. V. Šverák, Examples of rank–one convex functions. Proc. R. Soc. Edinb. A 114 (1990) 237–242. [Google Scholar]
  76. V. Šverák, Quasiconvex functions with subquadratic growth. Proc. R. Soc. Lond. A 433 (1991), 723–725. [Google Scholar]
  77. V. Šverák, Rank–one convexity does not imply quasiconvexity. Proc. R. Soc. Edinb. 120 (1992) 185–189. [Google Scholar]
  78. V. Šverák, Lower semicontinuity of variational integrals and compensated compactness, in Proc. of the Internaional Congress of Mathematicians, Zürich, Switzerland 1994, Birkhäuser Verlag, Basel, Switzerland (1995) 1153–1158. [Google Scholar]
  79. V. Šverák, On the problem of two wells, in Microstructures and phase transitions, D. Kinderlehrer, R.D. James, M. Luskin and J. Ericksen Eds. Springer, IMA Vol. Appl. Math. 54 (1993) 183–189. [Google Scholar]
  80. L. Tartar, Compensated compactness and applications to partial differential equations. Nonlinear Analysis and Mechanics: Heriot–Watt Symp., Vol. IV, R. Knops Ed. Pitman Res. Notes Math. 39 (1979) 136–212. [Google Scholar]
  81. L. Tartar, The compensated compactness method applied to systems of conservation laws. Systems of Nonlinear Partial Differential Eq., J.M. Ball Ed. Reidel (1983) 263–285. [Google Scholar]
  82. L. Tartar, Some remarks on separately convex functions, Microstructure and Phase Transitions, D. Kinderlehrer, R.D. James, M. Luskin and J.L. Ericksen Eds. Springer, IMA Vol. Math. Appl. 54 (1993) 191–204. [Google Scholar]
  83. B. Yan, On rank–one convex and polyconvex conformal energy functions with slow growth. Proc. R. Soc. Edinb. 127 (1997) 651–663. [Google Scholar]
  84. K.W. Zhang, A construction of quasiconvex functions with linear growth at infinity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. Ser. IV XIX (1992) 313–326. [Google Scholar]
  85. K.W. Zhang, Biting theorems for Jacobians and their applications. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 7 (1990) 345–365. [Google Scholar]
  86. K.W. Zhang, On various semiconvex hulls in the calculus of variations. Calc. Var. Partial Differ. Equ. 6 (1998) 143–160. [Google Scholar]

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