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Characterizations of young measures generated by gradients

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References

  1. Acerbi, E. & Fusco, N., 1984, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., 86, 125–145.

    Google Scholar 

  2. Balder, E. J., 1984, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Opt., 22, 570–597.

    Google Scholar 

  3. Ball, J. M., 1984, Singular minimizers and their significance in elasticity, Phase Transformations and Material Instabilities in Solids, (Gurtin, M., ed.) Academic Press, 1–20.

  4. Ball, J. M., 1989, A version of the fundamental theorem for Young measures, PDE's and continuum models of phase transitions, Lecture Notes in Physics, 344, (Rascle, M., Serre, D., & Slemrod, M., eds.) Springer, 207–215.

  5. Ball, J. M., 1990, Sets of gradients with no rank-one connections, J. math pures et appl., 69, 241–259.

    Google Scholar 

  6. Ball, J. M. & James, R., 1987, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100, 15–52.

    Google Scholar 

  7. Ball, J. M. & James, R., 1991, Proposed experimental tests of a theory of fine microstructure and the two well problem (to appear).

  8. Ball, J. M. & Murat F., 1984, W 1,p-quasiconvexity and variational problems for multiple integrals, J. Anal., 58, 225–253.

    Google Scholar 

  9. Ball, J. M. & Murat, F., 1989, Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc., 107, 655–663.

    Google Scholar 

  10. Ball, J. M. & Murat, F., Remarks on rank-one convexity and quasiconvexity, to appear.

  11. Ball, J. M. & Zhang, K., 1990, Lower semicontinuity of multiple integrals and the biting lemma, Proc. Royal Soc. Edinburgh, 114A, 367–379.

    Google Scholar 

  12. Battacharya, K., Self accomodation in martensite.

  13. Battacharya, K., Wedge-like microstructure in martensite.

  14. Berliocchi, H. & Lasry, J. M., 1973, Integrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France, 101, 129–184.

    Google Scholar 

  15. Brandon, D. & Rogers, R., Nonlocal regularization of L. C. Young's tacking problem, Appl. Math. Opt. (to appear).

  16. Capuzzo Dolcetta, I. & Ishii, H., 1984, Approximate solution of the Bellman equation of deterministic control theory, Appl. Math. Opt., 102, 161–181.

    Google Scholar 

  17. Chipot, M. & Collins, C., Numerical approximation in variational problems with potential wells, (to appear).

  18. Chipot, M. & Kinderlehrer, D., 1988, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103, 237–277.

    Google Scholar 

  19. Chipot, M., Numerical analysis of oscillations in nonconvex problems, in preparation.

  20. Chipot, M., Collins, C., & Kinderlehrer, D., Numerical analysis of oscillations in multiple well problems, in preparation.

  21. Chipot, M., Kinderlehrer, D., & Vergara-Caffarelli, G., 1986, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96, 81–96.

    Google Scholar 

  22. Collins, C. & Luskin, M., 1989. The computation of the austenitic-martensitic phase transition, PDE's and continuum models of phase transitions (Rascle, M., Serre, D., & Slemrod, M., eds.), Springer Lecture Notes in Physics, 344, 34–50.

  23. Collins, C. & Luskin, M., Numerical modeling of the microstructure of crystals with symmetry-related variants, Proc. ARO US-Japan Workshop on Smart/Intelligent Materials and Systems, Technomic.

  24. Collins, C. & Luskin, M., Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem (to appear).

  25. Collins, C., Kinderlehrer, D., & Luskin, M., 1991, Numerical approximation of the solution of a variational problem with a double well potential, SIAM J. Numer. Anal. 28, 321–322.

    Google Scholar 

  26. Dacorogna, B., 1982, Weak continuity and weak lower semicontinuity of non-linear functionals, Springer Lecture Notes 922 (1982).

  27. Dacorogna, B., 1989, Direct methods in the Calculus of Variations, Springer.

  28. Ericksen, J. L., 1979, On the symmetry of deformable crystals, Arch. Rational Mech. Anal., 72, 1–13.

    Google Scholar 

  29. Ericksen, J. L., 1980, Some phase transitions in crystals, Arch. Rational Mech. Anal., 73, 99–124.

    Google Scholar 

  30. Ericksen, J. L., 1981, Changes in symmetry in elastic crystals, IUTAM Symp. Finite Elasticity (Carlson, D. E. & Shield, R. T., eds.) M. Nijhoff, 167–177.

  31. Ericksen, J. L., 1981, Some simpler cases of the Gibbs phenomenon for thermoelastic solids, J. Thermal Stress, 4, 13–30.

    Google Scholar 

  32. Ericksen, J. L., 1982, Crystal lattices and sublattices, Rend. Sem. Mat. Padova, 68, 1–9.

    Google Scholar 

  33. Ericksen, J. L., 1983, Ill posed problems in thermoelasticity theory, Systems of Nonlinear Partial Differential Equations (Ball, J., ed.), D. Reidel, 71–95.

  34. Ericksen, J. L., 1984, The Cauchy and Born hypotheses for crystals, Phase Transformations and Material Instabilities in Solids (Gurtin, M., ed), Academic Press, 61–78.

  35. Ericksen, J. L., 1986, Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures, 22, 951–964.

    Google Scholar 

  36. Ericksen, J. L., 1986, Stable equilibrium configurations of elastic crystals, Arch. Rational Mech. Anal. 94, 1–14.

    Google Scholar 

  37. Ericksen, J. L., 1987, Twinning of crystals I, Metastability and Incompletely Posed Problems, IMA Vol. Math. Appl. 3, (Antman, S., Ericksen, J. L., Kinderlehrer, D., Müller, I., eds) Springer, 77–96.

  38. Ericksen, J. L., 1988, Some constrained elastic crystals, Material Instabilities in Continuum Mechanics, (Ball, J., ed.) Oxford, 119–136.

  39. Ericksen, J. L., 1989, Weak martensitic transformations in Bravais lattices, Arch. Rational Mech. Anal., 107, 23–36.

    Google Scholar 

  40. Evans, L. C., 1990, Weak convergence methods for nonlinear partial differential equations, C.B.M.S. 74, Amer. Math. Soc.

  41. Fonseca, I., 1985, Variational methods for elastic crystals, Arch. Rational Mech. Anal., 97, 189–220.

    Google Scholar 

  42. Fonseca, I., 1988, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. pures appl., 67, 175–195.

    Google Scholar 

  43. Fonseca, I., Lower semicontinuity of surface measures (to appear).

  44. Fonseca, I, The Wulff Theorem revisited (to appear).

  45. James, R. D., 1988, Microstructure and weak convergence, Proc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, (Ball, J. M., ed.), Oxford, 175–196.

  46. James, R. D. & Kinderlehrer, D., 1989, Theory of diffusionless phase transitions, PDE's and continuum models of phase transitions, Lecture Notes in Physics, 344, (Rascle, M., Serre, D., & Slemrod, M., eds.) Springer, 51–84.

  47. James, R. D. & Kinderlehrer, D., 1990, Frustration in ferromagnetic materials, Cont. Mech. Therm. 2, 215–239.

    Google Scholar 

  48. James, R. D. & Kinderlehrer, D., A theory of magnetostriction with application to TbDyFe2 (to appear).

  49. Kinderlehrer, D., 1988, Remarks about the equilibrium configurations of crystals, Young Measures Generated by Gradients 365 Proc. Symp. Material instabilities in continuum mechanics, Heriot-Watt (Ball, J. M. ed.) Oxford, 217–242.

  50. Kinderlehrer, D. & Pedregal, P., Charactérisation des mesures de Young associées à un gradient, C. R. Acad. Sci. Paris (to appear).

  51. Kinderlehrer, D. & Pedregal, P., Remarks about Young measures supported on two wells.

  52. Kinderlehrer, D. & Pedregal, P., Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. (to appear).

  53. Kinderlehrer, D. & Vergara-Caffarelli, G., 1989, The relaxation of functionals with surface energies, Asymptotic Analysis 2, 279–298.

    Google Scholar 

  54. Kohn, R. V., The relaxation of a double-well energy, Cont. Mech. Therm. (to appear).

  55. Liu, F.-C., 1977, A Luzin type property of Sobolev functions, Ind. Univ. Math. J., 26, 645–651.

    Google Scholar 

  56. Matos, J., The absence of fine microstructure in α-β quartz.

  57. Matos, J., Thesis, University of Minnesota.

  58. Morrey, C. B., Jr., 1966, Multiple Integrals in the Calculus of Variations, Springer.

  59. Murat, F., 1978, Compacité par compensation, Ann. Scuola Norm. Pisa, 5, 489–507.

    Google Scholar 

  60. Murat, F., 1979, Compacité par compensation II, Proc. int. meeting on recent methods in nonlinear analysis, Pitagora, 245–256.

  61. Murat, F., 1981, Compacité par compensation III, Ann. Scuola Norm. Pisa, 8, 69–102.

    Google Scholar 

  62. Pedregal, P., 1989, Thesis, University of Minnesota.

  63. Pedregal, P., 1989, Weak continuity and weak lower semicontinuity for some compensation operators, Proc. Royal Soc. Edin. 113, 267–279.

    Google Scholar 

  64. Sverak, V., Quasiconvex functions with subquadratic growth (to appear).

  65. Tartar, L., 1979, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot Watt Symposium, Vol. IV (Knops, R., ed.) Pitman Res. Notes in Math. 39, 136–212.

  66. Tartar, L., 1983, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Ball, J. M., ed.), Riedel.

  67. Tartar, L., 1984, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer Lecture Notes in Physics, 195, 384–412.

    Google Scholar 

  68. Warga, J., 1972, Optimal control of differential and functional equations, Academic Press.

  69. Young, L. C., 1969, Lectures on calculus of variations and optimal control theory, Saunders.

  70. Zhang, K., 1990, Biting theorems for Jacobians and their applications, Anal. Non-linéaire, 7, 345–366.

    Google Scholar 

  71. Zhang, K., A construction of quasiconvex functions with linear growth at infinity (to appear).

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Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, 15213, Pittsburgh, Pennsylvania

    David Kinderlehrer & Pablo Pedregal

  2. Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040, Madrid

    David Kinderlehrer & Pablo Pedregal

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  1. David Kinderlehrer
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Kinderlehrer, D., Pedregal, P. Characterizations of young measures generated by gradients. Arch. Rational Mech. Anal. 115, 329–365 (1991). https://doi.org/10.1007/BF00375279

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