Miary Younga i ich zastosowania w mikromechanice i optymalizacji. Część I. Podstawy matematyczne

• Autorzy: Bielski W. , Kruglenko E. , Telega J. J.
• Języki publikacji: PL

Słowa kluczowe

PL

mikromechanika ośrodków ciągłych; modelowanie matematyczne; miara Younga; H-miary

• Wydawca: Polskie Towarzystwo Matematyczne
• Czasopismo: Matematyka Stosowana : matematyka dla społeczeństwa
• Rocznik: 2003
• Tom: Nr 4
• Strony: 90--138
• Opis fizyczny: Bibliogr. 175 poz., wykr.

Twórcy

autor

Bielski W.

• Instytut Geofizyki, Polska Akademia Nauk, ul. Księcia Janusza 64, 01-452 Warszawa, Polska

autor

Kruglenko E.

• Instytut Podstawowych Problemów Techniki, Polska Akademia Nauk, ul. Świętokrzyska 21, 00-049 Warszawa, Polska

autor

Telega J. J.

• Instytut Podstawowych Problemów Techniki, Polska Akademia Nauk, ul. Świętokrzyska 21, 00-049 Warszawa, Polska

Bibliografia

• [1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal. 86 (1984), 125-145.
• [2] G. Alberti, S. Müller, A new approach to variational problems with multiple scales, Comm. Pure Appl. Math. 54 (2001), 761-825.
• [3] G. Alberti, G. Bouchitte, P. Seppecher, Phase transition with the line-tension effect, Arch. Rat. Mech. Anal. 144 (1998), 1-46.
• [4] J. J. Alibert, G. Bouchitte, Non uniform integrability and generalized Young measures, J. Convex Anal. 4 (1997), 129-147.
• [5] G. Allaire, Homogenization and Applications to Material Sciences, Lecture Notes at the Newton Inst., Cambridge, 1999.
• [6] G. Allaire, H. Maillot, H-measures and bounds on the effective properties of composite materials, Portugaliae Math. 60 (2003), 161-192.
• [7] N. Antonic, H-measures in hyperbolic symmetric systems, Proc. Roy. Soc. Edinburgh 126A (1996), 1133-1155.
• [8] N. Antonic, N. Balenovic, Homogenization and optimal design for plates, ZAMM 80 (2000), S757-S758.
• [9] G. Aubert, R. Tahraoui, Young measures, relaxation of functionals and existence results without weak lower semicontinuity, Adv. Diff. Equations 3, 1998.
• [10] E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control Optim. 22 (1984), 570-597.
• [11] E. J. Balder, Young measures techniques for existence of Carnot-Nash-Waltras equilibria, w: Topics in Mathematical Economics and Game Theory, M. Wooders (red.), Fields Institute Communications 23, Amer. Math. Soc., Providence, 1999, 31-39.
• [12] E. J. Balder, Lectures on Young measures theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste 31 (2000), suppl. 1, 1-69.
• [13] E. J. Balder, C. Hess, Two generalizations of Komlos’ theorem with lower closure-type applications, J. Convex Anal. 3 (1996), 25-44.
• [14] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal. 63 (1977), 337-403.
• [15] J. M. Ball, A version on the fundamental theorem for Young measures, w: Partial Differential Equations and Continuum Models of Phase Transitions, Springer, Berlin, 1989.
• [16] J. M. Ball, On the calculus of variations and sequentially weakly continuous maps, w: Ordinary and Partial Differential Equations (Dundee, 1976), Lecture Notes in Math., Springer, Berlin, 1976.
• [17] J. M. Ball, R. D. James, Fine phase mixtures as minimizers of energy, Arch. Rat. Mech. Anal. 100 (1987), 13-52.
• [18] J. M. Ball, R. D. James, Proposed experimental tests of a theory of fine microstructure and two-well problem, Phil. Trans. Roy. Soc. London A 338 (1992), 389-450.
• [19] J. M. Ball, F. Murat, Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc. 107 (1989), 655-663.
• [20] J. M. Ball, K. Zhang, Lower semicontinuity of multiple integrals and the Biting Lemma, Proc. Roy. Soc. Edinburgh 114A (1990), 367-379.
• [21] J. M. Ball, J. Currie, P. Olver, Null lagrangians, weak continuity, and variational problems of arbitrary order, J. Funct. Anal. 421 (1981), 315-328.
• [22] J. M. Ball, B. Kirchheim, J. Kristensen, Regularity of quasiconvex envelopes, Calc. Var. 11 (2000), 333-359.
• [23] A. Blake, A. Zisserman, Visual Reconstruction, MIT Press, Cambridge 1987.
• [24] P. Buelik, M. Luskin, Stability of microstructure for tetragonal to monoclinic martensitic transformations, Math. Model. Numer. Anal. 34 (2000), 663-685.
• [25] H. Berliocchi, J-M. Lasry, Intégrandes normales et mesures parametrées en calcul des variations, Bull. Soc. Math. France 101 (1973), 129-184.
• [26] K. Bhattahyara, R. James, A theory of thin films of martensitic materials with applications to microactuators, J. Mech. Phys. Solids 47 (1999), 531-576.
• [27] K. Bhattahyara, B. Li, M. Luskin, Uniqueness and stability of the simply laminated microstructure for martensic crystals that undergo a cubic to orthorombic phase transformation, preprint, 1998.
• [28] K. Bhattahyara, B. Li, M. Luskin, The simply laminated microstructure in martensic crystals that undergo a cubic-to-orthorombic phase transformation, Arch. Rat. Mech. Anal. 149 (1999), 123-154.
• [29] K. Bhattahyara, N. Firoozye, R. D. James, R. V. Kohn, Restrictions on microstructure, Proc. Roy. Soc. Edinburgh 124A (1994), 843-878.
• [30] W. Bielski, E. Kruglenko, J. J. Telega, Nonconvex minimization problems and microstructure, Szczęśniak (red.), Transactions, 2002.
• [31] M. Bocea, I. Fonseca, Equi-integrability results for 3D-2D dimension reduction problems, ESAIM Control Optim. Calc. Var. 7 (2002), 443.
• [32] E. Bonnetier, C. Conca, Approximation of Young measures by functions and applications to a problem of optimal design for plates with variable thickness, Proc. Roy. Soc. Edinburgh 124A (1994), 399-422.
• [33] A. Braides, T-convergence for Beginners, Oxford University Press, 2002.
• [34] A. Braides, I. Fonseca, G. Leoni, A-quasiconvexity: relaxation and homogenization, ESAIM Control Optim. Calc. Var. 5 (2000), 58.
• [35] D. Brandon, R. C. Rogers, Nonlocal regularization of L. C. Young’s tacking problem, Appl. Math. Optim. 25 (1992), 287-301.
• [36] G. Buttazzo, Semicontinuity, Relaxation and Integral Representation Problem in the Calculus of Variations, Pitman Res. Notes in Math. 207, Longman, Harlow, 1989.
• [37] C. Carstensen, Numerical Analysis of Microstructure, Lecture Note 10/2001, Max Planck Inst. Math. Sci. 2001.
• [38] C. Carstensen, A. Prohl, Numerical analysis of relaxed micromagnetics by penalised finite element, Numer. Math. 2000.
• [39] C. Carstensen, K. Hackl, On microstructure occuring in a model of finite-strain elastoplasticity involving a single slip-system, Z. Angew. Math. Mech. 80 (2000), S421-422.
• [40] C. Carstensen, A. Prohl, Numerical analysis of a relaxed model in micromagnetics, Z. Angew. Math. Mech. 80 (2000), S821-S822.
• [41] C. Carstensen, A. Prohl, Numerical analysis of relaxed micromagnetics by penalised finite elements, Numer. Math. 90 (2001), 65-99.
• [42] J. Chabrowski, Z. Kewei, On variational approach to photometric stereo, Ann. Inst. H. Poincaré 10 (1993), 363-375.
• [43] M. Chipot, Hyperelasticity for crystals, Euro. J. Appl. Math. 1 (1990), 113-129.
• [44] M. Chipot, Approximation and oscillations, w: Microstructure and Phase Transition, D. Kinderlehrer, R. James, M. Luskin, J. L. Ericksen (red.), Springer, Berlin 1993.
• [45] M. Chipot, V. Lécuyer, Some microstructures in three dimensions, w: Numerical methods for viscosity solutions and applications (Heraklion, 1999), Ser. Adv. Math. Appl. Sci., 59, World Sci., River Edge, NJ, 2001, 59-76.
• [46] M. Chipot, D. Kinderlehrer, G. Vergara-Caffarelli, Smoothness of linear laminates, Arch. Rat. Mech. Anal. 96 (1986).
• [47] P. G. Ciarlet, Mathematical Elasticity. Volume I: Three Dimensional Elasticity, North-Holland, Amsterdam, 1988.
• [48] S. Conti, A. DeSimone, G. Dolzmann, S. Müller, F. Otto, Multiscale modeling of materials - the role of analysis, preprint, 2002.
• [49] B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals, Lecture Notes in Math. 922, Springer, Berlin, 1982.
• [50] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer, Berlin, 1989.
• [51] B. Dacorogna, J.-P. Haeberly, Some numerical methods for the study of the convexity notions arising in the calculus of variations, EPFL Technical Report no. 06.95, July, 1995.
• [52] R. J. DiPerna, Convergence of approximate solutions to conservative laws, Arch. Rat. Mech. Anal. 82 (1983), 27-70.
• [53] R. J. DiPerna, Compensated compactness and general systems of conservative laws, Trans. Amer. Math. Soc. 292 (1985), 383-442.
• [54] R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88 (1985), 223-270.
• [55] R. J. DiPerna, A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), 667-689.
• [56] W. E, Homogenization of linear and nonlinear transport equations, Comm. Pure Appl. Math. 45 (1992), 301-326.
• [57] Y. Efendiev, M. Luskin, Stability of microstructures for some martensitic transformations, Math. Comput. Modelling 34 (2001), 1289-1305.
• [58] J. J. Egozcue, R. Meziat, P. Pedregal, From a nonlinear, nonconvex variational problem to a linear, convex formulation, Appl. Math. Optim. 47 (2002), 27-44.
• [59] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS 74, Amer. Math. Soc., 1990.
• [60] I. Fonseca, G. Parry, Variational problems for crystals with defects, w: Microstructure and Phase Transition, D. Kinderlehrer, R. James, M. Luskin, J. L. Ericksen (red.), Springer, Berlin, 1993.
• [61] I. Fonseca, D. Kinderlehrer, P. Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. Partial Differential Equations 2 (1994), 283-313.
• [62] I. Fonseca, Higher order variational problems, in: 2001 Centre of Nonlinear Analysis, Summer School, May 30-June 9, 2001.
• [63] I. Fonseca, G. Leoni, Relaxation results in micromagnetics, Ricerche Mat. 49 (2000), 269.
• [64] I. Fonseca, S. Maly, Relaxation of multiple integrals below the growth exponent, Ann. Inst. H. Poincaré14 (1997), 309-338.
• [65] I. Fonseca, S. Müller, Quasiconvex integrands and lower semicontinuity in L1, SIAM J. Math. Anal. 23 (1992), 1081-1098.
• [66] I. Fonseca, S. Müller, P. Pedregal, Analysis of concentration and oscillation effects generated by gradients, SIAM J. Math. Anal. 29 (1998), 736-756.
• [67] G. A. Francfort, F. Murat, Oscillations and energy densities in the wave equation, Comm. Part. Diff. Eq. 17 (1992), 1785-1865.
• [68] P. Gerard, Mesures semi-classiques et ondes de Bloch, Séminaire EDP 1990-1991, Ecole Polytechnique, Palaiseau, 1991.
• [69] P. Gerard, Microlocal defect measure, Comm. Part. Diff. Eq. 16 (1991), 1761-11794.
• [70] P. Gerard, P. A. Markowich, N. J. Mauser, F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (1997), 323-379.
• [71] M. K. Gobbert, A. Prohl, A comparison of classical and new finite element methods for the computation of laminate microstructure, Appl. Numer. Math. 36 (2001), 155-178.
• [72] S. Govindjee, A. Mielke, G. A. Hall, The free-energy of mixing for n-variant martensic phase transformations using quasi-convex analysis, J. Mech. Phys. Solids 50 (2002), 1897-1922.
• [73] M. Guidorzi, L. Poggiolini, Lower semicontinuity for quasiconvex integrals of higher order, Nonlinear Diff. Eq. A 6 (1999), 227-246.
• [74] D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc. 37 (2000), 407-436.
• [75] L. Hörmander, The Analysis of Partial Differential Operators III, Springer, Berlin 1985.
• [76] N. Hungerbühler, A refinement of Ball’s theorem on Young measures, New York J. Math. 3 (1997), 48-53.
• [77] N. Hungerbühler, Quasilinear elliptic systems in divergence form with weak monotonocity, New York J. Math. 5 (1999), 83-90.
• [78] N. Hungerbühler, Young measures and nonlinear PDEs, habilitation thesis, Birmingham, October 1999.
• [79] R. James, D. Kinderlehrer, Theory of diffusionless phase transitions, w: PDE’s and Continuum Models of Phase Transitions, M. Rascle, D. Serre, M. Slemrod (red.), Lecture Notes in Physics 344, Berlin, 1989.
• [80] A. Kałamajska, On lower semicontinuity of multiple integrals, Colloq. Math. 74 (1997).
• [81] A. Kałamajska, Between the classical theorem of Young and convergence theorem in set-valued analysis, preprint no. 633, Instytut Matematyki Polskiej Akademii Nauk, 2003.
• [82] D. Kinderlehrer, P. Pedregal, Characterization of Young measures generated by gradients, Arch. Rat. Mech. Anal. 115 (1991), 329-365.
• [83] D. Kinderlehrer, P. Pedregal, Remarks about Young measures supported on two wells, (1996), 329-365.
• [84] D. Kinderlehrer, P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. 23 (1992), 1-19.
• [85] D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. 4 (1994).
• [86] D. Kinderlehrer, D. E. Mason, Incoherence at heterogeneous interfaces, J. Mech. Physics Solids 47 (1999), 1609-1632.
• [87] R. V. Kohn, The relaxation of double-well energy, Cont. Mech. Thermodyn. 3 (1991), 193-236.
• [88] R. V. Kohn, F. Otto, Small surface energy, coarse-graining, and selection of microstructure, Physica D 107 (1997), 272-289.
• [89] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, Math. Inst., Tech. Univ. of Denmark, Mat-Report No. 1994-34, 1994.
• [90] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions, Math. Ann. 313 (1999), 653-710.
• [91] M. Kružik, T. Roubiček, On the measures of DiPerna a and Majda, Math. Bohemica 122 (1997), 383-399.
• [92] M. Kružik, T. Roubiček, Weierstrass-type maximum principle for microstructure in micromagnetics, Z. Anal. Anwend. 19 (2000), 415-428.
• [93] M. Kružik, T. Roubiček, Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim. 20 (1999), 511-530.
• [94] M. Kružik, A. Prohl, Approximation of Young measures in micromagnetic problems, preprint.
• [95] P. H. Leo, S. Müller, H-measures for splitting particles, preprint, 2000.
• [96] T. Lewiński, J. J. Telega, Plates, Laminates and Shells: Asymptotic Analysis and Homogenization, World Sci., Singapore, 2000.
• [97] G. Letta, Semicontinuité de certaines fonctionnelles sur l’espace des mesures de Young, Rend. Accad. Naz. Sci. 20 (1996), 215-219.
• [98] B. Li, M. Luskin, Approximation of a martensitic laminate with varying volume fractions, Math. Modelling Numer. Anal. 33 (1999), 67-87.
• [99] B. Li, Approximation of martensitic microstructure with general homogeneous boundary data, J. Math. Anal. Appl. 266 (2002), 451-467.
• [100] Z. Li, A mesh transformation method for computing microstructures, Numer. Math. 89 (2001), 511-533.
• [101] Z. Li, Relaxation results in micromagnetics. Computations of needle-like microstructures. Appl. Numer. Math. 39 (2001), 1-15.
• [102] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris 1969.
• [103] M. Luskin, Numerical analysis of a microstructure for a rotationally invariant, double well energy, ZAMM, J. Angew. Math. Mech. 76 (1996), S2-S4.
• [104] M. Luskin, Approximation of a laminated microstructure for a rotationally invariant, double well energy density, Numer. Math. 75 (1996), 205-221.
• [105] B. Li, M. Luskin, Theory and computation for the microstructure near the interface between twinned layers and a pure variant of martensite, Mat. Sci. Engn. A 273-275 (1999), 237-240.
• [106] Z. P. Li, Lower semicontinuity of multiple integrals and convergent integrands, ESAIM Control Optim. Calc. Var. 1 (1996), 169-189.
• [107] J. Málek, J. Nečas, M. Rokyta, M. Růžička, Weak and Measure-valued Solutions to Evolutionary PDEs, Chapman and Hall, London 1996.
• [108] J. Málek, J. Nečas, M. Rokyta, M. Růžička, On the non-newtonian incompressible fluids, Math. Models Meth. Appl. Sci. 3 (1993), 35-63.
• [109] J. Matos, Young measures and the absence of fine microstructures in a class of phase transition, Eur. J. Appl. Math. 3 (1992), 31-54.
• [110] E. J. McShane, Generalized curves, Duke Math. J. 6 (1940), 513-536.
• [111] A. Mielke, Flow properties for Young-measure solutions of semilinear hyperbolic problems, Proc. Roy. Soc. Edinburgh 129 A (1999), 85-123.
• [112] C. B. Morrey, Quasiconvexity and the lowersemicontinuity of multiple integrals, Pac. J. Math. 2 (1952), 25-53.
• [113] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer, Berlin, 1965.
• [114] S. Müller, Variational models for microstructure and phase transitions, w: Calculus of Variations and Geometric Evolution Problems, S. Hildebrandt, M. Struwe (red.), Lecture Notes in Math. 1713, Springer, 1999, 85-210.
• [115] S. Müller, I. Fonseca, A-quasiconvexity, lower semicontinuity and Young measures, SIAM J. Math. Anal. (1998).
• [116] J. Muñoz, P. Pedregal, Equilibrium conditions for Young measures.
• [117] J. Muñoz, P. Pedregal, Explicit solutions of nonconvex variational problems in dimension one, Appl. Math. Optim. 41 (2000), 129-140.
• [118] R. A. Nicolades, N. Walkington, H. Wang, Numerical methods for a nonconvex optimization problem modeling martensitic microstructure, SIAM J. Sci. Comput. 4 (1993), 1122-1141.
• [119] R. A. Nicolaides, N. J. Walkington, Computation of microstructure utilizing Young measure representations, J. Intelligent Materials Systems and Structures 4 (1993), 795-800.
• [120] P. Pedregal, Laminates and microstructure, Eur. J. Appl. Math. 4 (1992), 121-149.
• [121] P. Pedregal, Relaxation in ferromagnetism: The rigid case, J. Nonlinear Sci. 4 (1994), 105-125.
• [122] P. Pedregal, The concept of laminate, w: Microstructure and Phase Transition, D. Kinderlehrer, R. James, M. Luskin, J. L. Ericksen (red.), Springer, Berlin, 1993.
• [123] P. Pedregal, Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997.
• [124] P. Pedregal, Equilibrium conditions for Young measures, SIAM J. Control. Optim. 36 (1998), 797-813.
• [125] P. Pedregal, Numerical approximation of parametrized measures, Numer. Funct. Anal. Optim. 16 (1995), 1049-1066.
• [126] P. Pedregal, Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32 (2000), 854-869.
• [127] P. Pedregal, Relaxation in magnetostriction, Calc. Var. Partial Differential Equations 10 (2000), 1-19.
• [128] P. Pedregal, On the numerical analysis of non-convex variational problem, Numer. Math. 74 (1996), 325-336.
• [129] P. Pedregal, Optimization, relaxation and Young measure, Bull. Amer. Math. Soc. 36 (1997), 27-58.
• [130] P. Pedregal, Constrained quasiconvexity and structural optimization, Arch. Rat. Mech. Anal. 154 (2000), 325-342.
• [131] P. Pedregal, Variational Methods in Nonlinear Elasticity, SIAM, 2000.
• [132] P. Pedregal, Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design, Electronic Res. Ann. AMS 7 (2001), 72-78.
• [133] R. Peszek, Oscillations of solutions to the two-dimensional Broadwell model, an H-measure approach, SIAM J. Math. Anal. 26 (1995), 750-760.
• [134] A. Prohl, Error analysis for the computation of microstructure in cubic ferromagnets, Math. Seminar, Christian-Albrechts-Univ. Kiel, preprint, 1999.
• [135] A. Prohl, An adaptive finite element method for solving a double well problem describing crystalline microstructure, Math. Model. Numer. Anal. 33 (1999), 781-796.
• [136] A. Prohl, An adaptive finite element method for solving a double well problem describing crystalline microstructure, Math. Model. Numer. Anal. 33 (1999), 781-796.
• [137] L. Piccini, M. Valadier, Uniform inteqrability and Young measures, J. Math. Anal. Appl. 195 (1995), 428-439.
• [138] A. Prohl, Computational Micromagnetism, Teubner, Stuttgart, 2001.
• [139] M. Rascle, On the ststic and dynamic study of oscillations for some nonlinear hyperbolic systems of conservation laws, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 333-350.
• [140] X. Ren, M. Winter, Young measures in a nonlocal phase transition problem, Proc. Roy. Soc. Edinburgh 127A (1997), 615-637.
• [141] M. O. Rieger, Young measure solutions for nonconvex elastodynamics, Max Planck Institute, Univ. Leipzig, preprint 62/2000.
• [142] T. Roubiček, Effective characterization of generalized Young measures generated by gradients, Boll. Un. Mat. Ital. 9-B (1995), 755-779.
• [143] T. Roubiček, Relaxation in Optimization Theory and Variational Calculus, de Gruyter, Berlin, 1997.
• [144] T. Roubiček, M. Kružik, Numerical treatment of microstructure evolution modelling, w: ENUMATH 97. Proceedings of the 2nd European conference on numerical mathematics and advanced applications (Heidelberg), H. G. Bock et al. (red.), World Sci., 1998, 532-539.
• [145] T. Roubiček, W. H. Schmidt, Existence in optimal control problems of certain Fredholm integral equations, Control Cybernetics 30 (2001), 303-322.
• [146] W. Rudin, Functional Analysis, McGraw-Hill, 1973.
• [147] M. Saadoune, M. Valadier, Extraction of a “good” subsequence from a bounded sequence of integrable functions, J. Convex Anal. 2 (1995), 345-357.
• [148] V. P. Smyshlayev, J. Willis, On the relaxation of a three-well energy, Proc. Roy. Soc. London A 455 (1988), 779-814.
• [149] V. Šverak, New examples of quasiconvex functions, Arch. Rat. Mech. Anal. 119 (1992), 293-300.
• [150] V. Šverak, On Tartar’s conjecture, Ann. Inst. H. Poincaré 10 (1993), 405-412.
• [151] M. A. Sychev, Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 755-782.
• [152] M. A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 773-812.
• [153] L. Tartar, Compensated compactness and applications to partial differential equations. w: Nonlinear Analysis and Mechanics: Heriot Watt Symposium, vol. IV, R. Knops (red.), Res. Notes in Math. 39, Pitman, London, 1979.
• [154] L. Tartar, The compensated compactness method applied to systems of conservation laws, in: Systems of Nonlinear Partial Differential Equations, J. M. Ball (red.), Reidel, 1983, 263-285.
• [155] L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh 115A (1990), 193-230.
• [156] L. Tartar, Etude des oscillation dans les équations aux derivées partielles nonlinéaires, Lecture Notes in Physics 195, 1984.
• [157] L. Tartar, H-measures and small amplitude homogenization, w: Composite Media, R. V. Kohn i G. Milton (red.), 1990, 89-99.
• [158] L. Tartar, Some remarks on separately convex functions, w: Microstructure and Phase Transition, D. Kinderlehrer, R. James, M. Luskin, J. L. Ericksen (red.), Springer, Berlin, 1993.
• [159] L. Tartar, Beyond Young measures, Meccanica 30 (1995), 505-526.
• [160] L. Tartar, Oscillations and concentrations effects in partial differential equations: Why waves may behave like particles, w: XVII CEDYA: Congress on Differential Equations and Applications/VII CMA: Congress on Applied Mathematics Salamanca, 2001), 179-219.
• [161] L. Tartar, An introduction to the homogenization method in optimal design, w: B. Kawohl, O. Pironneau, L. Tartar, J.-P. Zolezio, Optimal Shape Design, A. Cellina, A. Ornelas (red.), Lecture Notes in Math. 1740, Springer, Berlin, 2000.
• [162] L. Tartar, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures, w: Developments in Partial Differential Equations to Mathematical Physics, Buttazzo, Galdi, Zanghirati (red.), Plenum, New York, 1991.
• [163] L. Tartar, Homogenneisation et H-mesures, ESAIM: Actes du 30ème Congrès d’Analyse Numérique 6 (1998), 111-131.
• [164] J. J. Telega, T. Lewiński, On a saddle-point theorem in minimum compliance design, J. Optim. Theory Appl. 106 (2000), 441-450.
• [165] F. Theil, Young-measure solutions for a viscoelastically damped wave equation with nonmonotone stress-strain relation, Arch. Rat. Mech. Anal. 144 (1998), 47-78.
• [166] A. M. Toader, Memory effects and T-convergence: A time dependent case, J. Convex Anal. 6 (1999), 13-27.
• [167] A. M. Toader, Link between Young measures associated to constrained sequences, ESAIM Control Optim. Calc. Var. 5 (2000), 579-590.
• [168] M. Valadier, Young measures, w: Methods of Nonconvex Analysis, Lecture Notes in Math. 1446, Springer, Berlin, 1990.
• [169] M. Valadier, A course on Young measures, Rend. 1st. Mat. Trieste 26 (1994), 349-394.
• [170] M. Valadier, Admissible function in two-scale convergence, Portugal. Math. 54 (1997), 147-164.
• [171] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
• [172] L. C. Young, Generalized curves and the existence of an attained absolute minimum in the calculus of variations, C. R. Soc. Sci. Lettres de Varsovie Cl. III 30 (1937), 212-234.
• [173] L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, Chelsea, Philadelphia, 1969.
• [174] K. Zhang, Biting theorems for Jacobians and their applications, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), 345-366.
• [175] S. Zhou, A note on nonlinear elliptic systems involving measures, Electronic J. Diff. Equations 8 (2000), 1-6.