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Gradient Young measures generated by sequences in Sobolev spaces

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Abstract

Oscillatory properties of a weak convergent sequence of functions bounded inL p, 1 ≤p ≤ ∞, may be summarized by the parametrized measure it generates. When such a measure is generated by the gradients of a sequence of functions bounded inH 1,p, it must have special properties. The purpose of this paper is to characterize such parametrized measures as the ones that obey Jensen’s inequality for all quasiconvex functions with the appropriate growth at infinity. We have found subtle differences between the casesp < ∞ andp = ∞. A consequence is that any measure determined by biting convergence is in fact generated by a sequence convergent in a stronger sense. We also give a few applications.

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References

  1. Acerbi, E., and Fusco, N. Semicontinuity problems in the calculus of variations.Arch. Rat. Mech. Anal. 86, 125–145 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  2. Acerbi, E., and Fusco, N. An approximation lemma forW 1,p functions. In:Proc. Symp. Material Instabilities in Continuum Mechanics, edited by J. M. Ball, pp. 1–5. Oxford: Heriot-Watt 1988.

    Google Scholar 

  3. Ball, J. M. A version of the fundamental theorem for Young measures. InPDE’s and Continuum Models of Phase Transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 207–215. Springer.

  4. Ball, J. M. Sets of gradients with no rank-one connections.J. Math Pures et Appl. 69, 241–259 (1990).

    MATH  Google Scholar 

  5. Ball, J. M., and James, R. Fine phase mixtures as minimizers of energy.Arch. Rat. Mech. Anal. 100, 15–52 (1987).

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Ball, J. M., and Murat, F. Remarks on Chacon’s biting lemma.Proc. Am. Math. Soc. 107, 655–663 (1989).

    Article  MathSciNet  MATH  Google Scholar 

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

  10. Ball, J. M., and Zhang, K. Lower semicontinuity of multiple integrals and the biting lemma.Proc. R. Soc. Edinburgh 114A, 367–379 (1990).

    MathSciNet  Google Scholar 

  11. Battacharya, K. Wedge-like microstructure in martensite.Acta Metal. 39, 2431–2444 (1991).

    Article  Google Scholar 

  12. Battacharya, K. Self accomodation in martensite.Arch. Rat. Mech. Anal. 120, 201–244 (1992).

    Article  Google Scholar 

  13. Billingsley, P. 1986Probability and Measure. Wiley.

  14. Brandon, D., and Rogers, R. Nonlocal regularization of L. C. Young’s tacking problem (to appear).

  15. Brooks, J. K., and Chacon, R. V. Continuity and compactness of measures.Adv. in Math. 37, 16–26 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  16. Chipot, M. Numerical analysis of oscillations in nonconvex problems.Num. Math. 59, 747–767 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  17. Chipot, M., and Collins, C. Numerical approximation in variational problems with potential wells.SIAM J. Num. Anal. 29, 1002–1014 (1992).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Chipot, M., Kinderlehrer, D., and Vergara-Caffarelli, G. Smoothness of linear laminates.Arch. Rat. Mech. and Anal. 96, 81–96 (1986).

    MathSciNet  MATH  Google Scholar 

  20. Coifman, P.-L., Lions, Meyer, and Semmes, S. Compacité par compensation et espaces de Hardy.CRAS Paris 309, 945–949 (1989).

    MathSciNet  MATH  Google Scholar 

  21. Collins, C., and Luskin, M. The computation of the austenitic-martensitic phase transition. In:PDE’s and continuum models of phase transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 34–50. Springer 1989.

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

  23. Collins, C., and Luskin, M. Optimal order error estimates for the finite element approximation of the solution of a nonconvex variational problem.Math. Comp. 57, 621–637 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  24. Collins, C., Kinderlehrer, D., and Luskin, M. Numerical approximation of the solution of a variational problem with a double well potential.SIAM J. Numer. Anal. 28, 321–333 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  25. Dacorogna, B.Direct Methods in the Calculus of Variations. Springer 1989.

  26. Dal Maso, G. Integral representation onBV(Ω) of Γ-limits of variational integrals.Manuscripta Math. 30, 387–416 (1980).

    Article  MATH  Google Scholar 

  27. Ericksen, J. L. On the symmetry of deformable crystals.Arch. Rat. Mech. Anal. 72, 1–13 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  28. Ericksen, J. L. Some phase transitions in crystals.Arch. Rat. Mech. Anal. 73, 99–124 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  29. Ericksen, J. L. Changes in symmetry in elastic crystals.IUTAM Symp. Finite Elasticity, edited by D. E. Carlson and R. T. Shield, pp. 167–177. M. Nijhoff 1981.

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

    Article  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. Ericksen, J. L. Ill posed problems in thermoelasticity theory. InSystems of Nonlinear Partial Differential Equations, edited by J. Ball, pp. 71–95. D. Reidel 1983.

  33. Ericksen, J. L. The Cauchy and Born hypotheses for crystals. InPhase Transformations and Material Instabilities in Solids, edited by M. Gurtin, pp. 61–78. Academic Press 1984.

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Ericksen, J. L. Stable equilibrium configurations of elastic crystals.Arch. Rat. Mech. Anal. 94, 1–14 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  36. Ericksen, J. L. Twinning of crystals I. InMetastability and Incompletely Posed Problems, IMA Vol. Math. Appl., vol. 3, edited by S. Antman, J. L. Ericksen, D. Kinderlehrer, and I. M ller, pp. 77–96. Springer 1987.

  37. Ericksen, J. L. Some constrained elastic crystals. InMaterial Instabilities in Continuum Mechanics, edited by J. Ball, pp. 119–136. Oxford 1988.

  38. Ericksen, J. L. Weak martensitic transformations in Bravais lattices.Arch. Rat. Mech. Anal. 107, 23–36 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  39. Ericksen, J. L., James, R. D., Kinderlehrer, D., and Luskin, M. (eds.)Microstructure and Phase Transitions, IMA Volume 54. Springer 1993.

  40. Evans, L. C.Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74. American Mathematical Society 1990.

  41. Evans, L. C., and Gariepy, R. F. Blow-up, compactness, and partial regularity in the calculus of variations.Indiana Univ. Math. J. 36, 361–371 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  42. Evans, L. C., and Gariepy, R. F. Some remarks on quasiconvexity and strong convergence.Proc. R. Soc. Edinburgh Sect. A 106, 53–61 (1987).

    MathSciNet  MATH  Google Scholar 

  43. Firoozye, N., and Kohn, R. Geometric parameters and the relaxation of multiwell energies, IMA preprint Series 765 (1991).

  44. Fonseca, I. Variational methods for elastic crystals.Arch. Rat. Mech. Anal. 97, 189–220 (1985).

    MathSciNet  Google Scholar 

  45. Fonseca, I. The lower quasiconvex envelope of the stored energy function for an elastic crystal.J. Math. Pures et Appl. 67, 175–195 (1988).

    MathSciNet  MATH  Google Scholar 

  46. Fonseca, I. The Wulff Theorem revisited.Proc. R. Soc. Lond. 432, 125–145 (1991).

    MathSciNet  MATH  Google Scholar 

  47. Fonseca, I. Lower semicontinuity of surface energies.Proc. R. Acad. Edin. A 120, 99–115 (1992).

    MathSciNet  MATH  Google Scholar 

  48. Fonseca, I., and Müller, S. Relaxation of Quasiconvex functionals inB V(ό; ℝp) for Integrandsf(x, y, ∇u).Arch. Rat. Mech. Anal. 123, 1–51 (1993).

    Article  MATH  Google Scholar 

  49. Fonseca, I., and Müller, S. Quasiconvex integrands and lower semicontinuity inL 1.SIAM J. Math Anal. 23, 1081–1098 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  50. Iwaniec, T., and Sbordone, C. On the integrability of the Jacobian under minimal hypothesis.Arch. Rat. Mech. Anal. 119, 129–144 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  51. James, R. D. Microstructure and weak convergence. InProc. Symp. Material Instabilities in Continuum Mechanics,Heriot-Watt, edited by J. M. Ball, pp. 175–196. Oxford 1988.

  52. James, R. D., and Kinderlehrer, D. Theory of diffusionless phase transitions. InPDE’s and Continuum Models of Phase Transitions, Lecture Notes in Physics, vol. 344, edited by M. Rascle, D. Serre, and M. Slemrod, pp. 51–84. Springer 1989.

  53. James, R. D., and Kinderlehrer, D. Frustration in ferromagnetic materials.Cont. Mech. Therm. 2, 215–239 (1990).

    Article  MathSciNet  Google Scholar 

  54. James, R. D., and Kinderlehrer, D. A theory of magnetostriction with application to TbDyFe2.Phil. Mag. B 68, 237–274 (1993).

    Google Scholar 

  55. James, R. D., and Kinderlehrer, D. Frustration and microstructure: An example in magnetostriction.Proc. First Europ. Conf. Elliptic Parab. Problems. Pont a Mousson 1990.

  56. Kinderlehrer, D. Remarks about the equilibrium configurations of crystals. InProc. Symp. Material Instabilities in Continuum Mechanics, Heriot-Watt, edited by J. M. Ball, pp. 217–242. Oxford 1988.

  57. Kinderlehrer, D., and Pedregal, P. Characterizations of Young measures generated by gradients.Arch. Rat. Mech. Anal. 115, 329–365 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  58. Kinderlehrer, D., and Pedregal, P. Charactérisation des mesures de Young associées á un gradient.CRAS Paris 313, 765–770 (1991).

    MathSciNet  MATH  Google Scholar 

  59. Kinderlehrer, D., and Pedregal, P. Gradient Young measures generated by sequences in Sobolev spaces. InNonlinear PDE and their Applications, Collége de France Seminar. Longman (to appear).

  60. Kinderlehrer, D., and Pedregal, P. Remarks about the analysis of gradient Young measures. InPDE and Related Subjects, edited by M. Miranda. Bitman/Longman Res. Notes269, pp. 125–150 (1992).

  61. Kinderlehrer, D., and Pedregal, P. Weak convergence of integrands and the Young measure representation.SIAM J. Math. Anal. 23, 1–19 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  62. Kinderlehrer, D., and Pedregal, P. Remarks about Young measures supported on two wells (to appear).

  63. Kinderlehrer, D., and Vergara-Caffarelli, G. The relaxation of functionals with surface energies.Asymptotic Analysis 2, 279–298 (1989).

    MathSciNet  MATH  Google Scholar 

  64. Kohn, R. V. The relaxation of a double-well energy.Cont. Mech. Therm. 3, 193–236 (1991).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  66. Luskin, M., and Ma, L. Analysis of the finite element approximation of microstructure in micromagnetics. UMSI report 90/164 (1990).

  67. Matos, J. Thesis, University of Minnesota (1990).

  68. Matos, J. The absence of fine microstructure in α − β quartz.Eur. J. Appl. Math. 3, 31–54 (1992).

    Article  MathSciNet  MATH  Google Scholar 

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

  70. Müller, S. Weak continuity of determinants an nonlinear elasticity.CRAS Paris 307, 501–506 (1988).

    MATH  Google Scholar 

  71. Müller, S. A surprising higher integrability property of mappings with positive determinant.Bull. AMS 21, 245–248 (1989).

    Article  MATH  Google Scholar 

  72. Müller, S. Higher integrability of determinants and weak convergence inL 1.J. Reine Angew. Math. 412, 20–34 (1990).

    MathSciNet  MATH  Google Scholar 

  73. Nicolaides, R. A., and Walkington, N. Computation of microstructure utilizing Young measure representations.Proc. Recent Adv. Adaptive Sensory Materials Appl. (1990).

  74. Pedregal, P. Weak continuity and weak lower semicontinuity for some compensation operators.Proc. R. Soc. Edin. 113, 267–279 (1989).

    MathSciNet  MATH  Google Scholar 

  75. Pedregal, P. Laminates and microstructure.Eur. J. Appl. Math. 4, 121–149 (1993).

    MathSciNet  MATH  Google Scholar 

  76. Slemrod, M. Dynamics of measure valued solutions to a backward-forward heat equation.J. Dyn. Diff. Eqns. 3, 1–28 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  77. Sverak, V. On the regularity of the Monge-Ampére equation without convexity (to appear).

  78. Sverak, V. Quasiconvex functions with subquadratic growth.Proc. R. Soc. London A 433, 723–725 (1991).

    MathSciNet  MATH  Google Scholar 

  79. Sverak, V. Rank one convexity does not imply quasiconvexity.Proc. R. Soc. Edin. A 120, 185–189 (1992).

    MathSciNet  MATH  Google Scholar 

  80. Tartar, L. Compensated compactness and applications to partial differential equations. InNonlinear analysis and mechanics: Heriot Watt Symposium, vol. IV,Pitman Research Notes in Math., vol. 39, edited by R. Knops, pp. 136–212.

  81. Tartar, L. The compensated compactness method applied to systems of conservation laws. InSystems of Nonlinear Partial Differential Equations, edited by J. M. Ball. Riedel 1983.

  82. Tartar, L. Étude des oscillations dans les équations aux dérivées partielles nonlinéaires.Springer Lect. Notes Physics, vol. 195, pp. 384–412. Springer 1984.

  83. Young, L. C.Lectures on Calculus of Variations and Optimal Control Theory. W. B. Saunders 1969.

  84. Zhang, K. Biting theorems for Jacobians and their applications.Anal. Nonlineare 7, 345–366 (1990).

    MATH  Google Scholar 

  85. Zhang, K. A construction of quasiconvex functions with linear growth at infinity.Ann. S.N.S. Pisa 19, 313–326 (1992).

    MATH  Google Scholar 

  86. Zhang, K. Rank-one connections and the three “well” problem.Trans. AMS (to appear).

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Authors and Affiliations

  1. Department of Mathematics and Center for Nonlinear Analysis, Carnegie Mellon University, 15213-3890, Pittsburgh, PA, USA

    David Kinderlehrer & Pablo Pedregal

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

    David Kinderlehrer & Pablo Pedregal

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  1. David Kinderlehrer
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  2. Pablo Pedregal
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Research groupTransitions and Defects in Ordered Materials, funded by the NSF, the AFOSR, and the ARO. The work of the second author is also supported by DGICYT (Spain) through “Programa de Perfeccionamiento y Movilidad del Personal Investigador” and through Grant PB90-0245.

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Kinderlehrer, D., Pedregal, P. Gradient Young measures generated by sequences in Sobolev spaces. J Geom Anal 4, 59–90 (1994). https://doi.org/10.1007/BF02921593

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