×

zbMATH — the first resource for mathematics

Characterizations of Young measures generated by gradients. (English) Zbl 0754.49020
Il s’agit d’utiliser les méthodes variationelles pour étudier les configurations d’équilibre des solides cristallins. Pour cela la recherche présenté a comme point de départ la théorie de la thermoélasticité, qui a été l’objet de nombreuses publications récentes de J. L. Ericksen. Dans ce travail il est question des mesures paramétrisés ou mesures de Young, et avant tout on recherche quelles mesures ordinaires avec support en ensembles compactes de matrices peuvent dériver de limites de successions de gradients. Les auteurs donnent deux caractérisations, qui expriment la validité de l’inégalité de Jensen pour des classes de fonctions quasiconvexes (selon C. B. Morrey), l’une dans la classe des fonctions quasiconvexes qui sont identiquement \(+\infty\) au dehors d’une balle l’autre dans celle des fonctions quasiconvexes continues.
Reviewer: S.Cinquini (Pavia)

MSC:
90C52 Methods of reduced gradient type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Acerbi, E. & Fusco, N., 1984, Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., 86, 125-145. · Zbl 0565.49010 · doi:10.1007/BF00275731
[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. · Zbl 0549.49005 · doi:10.1137/0322035
[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. · Zbl 0991.49500
[5] Ball, J. M., 1990, Sets of gradients with no rank-one connections, J. math pures et appl., 69, 241-259. · Zbl 0644.49011
[6] Ball, J. M. & James, R., 1987, Fine phase mixtures as minimizers of energy, Arch. Rational Mech. Anal., 100, 15-52. · Zbl 0629.49020 · doi:10.1007/BF00281246
[7] Ball, J. M. & James, R., 1991, Proposed experimental tests of a theory of fine microstructure and the two well problem (to appear). · Zbl 0758.73009
[8] Ball, J. M. & Murat F., 1984, W 1,p -quasiconvexity and variational problems for multiple integrals, J. Anal., 58, 225-253. · Zbl 0549.46019
[9] Ball, J. M. & Murat, F., 1989, Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc., 107, 655-663. · Zbl 0678.46023
[10] Ball, J. M. & Murat, F., Remarks on rank-one convexity and quasiconvexity, to appear. · Zbl 0751.49005
[11] Ball, J. M. & Zhang, K., 1990, Lower semicontinuity of multiple integrals and the biting lemma, Proc. Royal Soc. Edinburgh, 114A, 367-379. · Zbl 0716.49011
[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. · Zbl 0282.49041
[15] Brandon, D. & Rogers, R., Nonlocal regularization of L. C. Young’s tacking problem, Appl. Math. Opt. (to appear). · Zbl 0769.49016
[16] Capuzzo Dolcetta, I. & Ishii, H., 1984, Approximate solution of the Bellman equation of deterministic control theory, Appl. Math. Opt., 102, 161-181. · Zbl 0553.49024 · doi:10.1007/BF01442176
[17] Chipot, M. & Collins, C., Numerical approximation in variational problems with potential wells, (to appear). · Zbl 0763.65049
[18] Chipot, M. & Kinderlehrer, D., 1988, Equilibrium configurations of crystals, Arch. Rational Mech. Anal., 103, 237-277. · Zbl 0673.73012 · doi:10.1007/BF00251759
[19] Chipot, M., Numerical analysis of oscillations in nonconvex problems, in preparation. · Zbl 0712.65063
[20] Chipot, M., Collins, C., & Kinderlehrer, D., Numerical analysis of oscillations in multiple well problems, in preparation. · Zbl 0824.65045
[21] Chipot, M., Kinderlehrer, D., & Vergara-Caffarelli, G., 1986, Smoothness of linear laminates, Arch. Rational Mech. Anal., 96, 81-96. · Zbl 0617.73062 · doi:10.1007/BF00251414
[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. · Zbl 0991.80502
[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). · Zbl 0735.65042
[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. · Zbl 0725.65067 · doi:10.1137/0728018
[26] Dacorogna, B., 1982, Weak continuity and weak lower semicontinuity of non-linear functionals, Springer Lecture Notes 922 (1982). · Zbl 0484.46041
[27] Dacorogna, B., 1989, Direct methods in the Calculus of Variations, Springer. · Zbl 0703.49001
[28] Ericksen, J. L., 1979, On the symmetry of deformable crystals, Arch. Rational Mech. Anal., 72, 1-13. · Zbl 0439.20031 · doi:10.1007/BF00250733
[29] Ericksen, J. L., 1980, Some phase transitions in crystals, Arch. Rational Mech. Anal., 73, 99-124. · Zbl 0429.73007 · doi:10.1007/BF00258233
[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. · doi:10.1080/01495738108909949
[32] Ericksen, J. L., 1982, Crystal lattices and sublattices, Rend. Sem. Mat. Padova, 68, 1-9. · Zbl 0523.73002
[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. · Zbl 0567.73112
[35] Ericksen, J. L., 1986, Constitutive theory for some constrained elastic crystals, Int. J. Solids Structures, 22, 951-964. · Zbl 0595.73001 · doi:10.1016/0020-7683(86)90030-2
[36] Ericksen, J. L., 1986, Stable equilibrium configurations of elastic crystals, Arch. Rational Mech. Anal. 94, 1-14. · Zbl 0597.73006 · doi:10.1007/BF00278240
[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. · Zbl 0655.73022
[39] Ericksen, J. L., 1989, Weak martensitic transformations in Bravais lattices, Arch. Rational Mech. Anal., 107, 23-36. · Zbl 0697.73005 · doi:10.1007/BF00251425
[40] Evans, L. C., 1990, Weak convergence methods for nonlinear partial differential equations, C.B.M.S. 74, Amer. Math. Soc. · Zbl 0698.35004
[41] Fonseca, I., 1985, Variational methods for elastic crystals, Arch. Rational Mech. Anal., 97, 189-220. · Zbl 0611.73023
[42] Fonseca, I., 1988, The lower quasiconvex envelope of the stored energy function for an elastic crystal, J. Math. pures appl., 67, 175-195. · Zbl 0718.73075
[43] Fonseca, I., Lower semicontinuity of surface measures (to appear). · Zbl 0757.49013
[44] Fonseca, I, The Wulff Theorem revisited (to appear). · Zbl 0725.49017
[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. · Zbl 0991.74504
[47] James, R. D. & Kinderlehrer, D., 1990, Frustration in ferromagnetic materials, Cont. Mech. Therm. 2, 215-239. · doi:10.1007/BF01129598
[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. · Zbl 0850.73037
[50] Kinderlehrer, D. & Pedregal, P., Charactérisation des mesures de Young associées à un gradient, C. R. Acad. Sci. Paris (to appear). · Zbl 0759.49005
[51] Kinderlehrer, D. & Pedregal, P., Remarks about Young measures supported on two wells. · Zbl 0833.49012
[52] Kinderlehrer, D. & Pedregal, P., Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. (to appear). · Zbl 0757.49014
[53] Kinderlehrer, D. & Vergara-Caffarelli, G., 1989, The relaxation of functionals with surface energies, Asymptotic Analysis 2, 279-298. · Zbl 0696.49007
[54] Kohn, R. V., The relaxation of a double-well energy, Cont. Mech. Therm. (to appear). · Zbl 0825.73029
[55] Liu, F.-C., 1977, A Luzin type property of Sobolev functions, Ind. Univ. Math. J., 26, 645-651. · Zbl 0368.46036 · doi:10.1512/iumj.1977.26.26051
[56] Matos, J., The absence of fine microstructure in ?-? quartz.
[57] Matos, J., Thesis, University of Minnesota.
[58] Morrey, C. B., Jr., 1966, Multiple Integrals in the Calculus of Variations, Springer. · Zbl 0142.38701
[59] Murat, F., 1978, Compacité par compensation, Ann. Scuola Norm. Pisa, 5, 489-507. · Zbl 0399.46022
[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. · Zbl 0464.46034
[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. · Zbl 0702.35054
[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. · Zbl 0536.35003
[67] Tartar, L., 1984, Étude des oscillations dans les équations aux dérivées partielles nonlinéaires. Springer Lecture Notes in Physics, 195, 384-412. · doi:10.1007/3-540-12916-2_68
[68] Warga, J., 1972, Optimal control of differential and functional equations, Academic Press. · Zbl 0253.49001
[69] Young, L. C., 1969, Lectures on calculus of variations and optimal control theory, Saunders. · Zbl 0177.37801
[70] Zhang, K., 1990, Biting theorems for Jacobians and their applications, Anal. Non-linéaire, 7, 345-366. · Zbl 0717.49012
[71] Zhang, K., A construction of quasiconvex functions with linear growth at infinity (to appear).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.