A new approach to Young measure theory, relaxation and convergence in energy
Sychev, M. A.
Annales de l'I.H.P. Analyse non linéaire, Tome 16 (1999), p. 773-812 / Harvested from Numdam
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     author = {Sychev, M. A.},
     title = {A new approach to Young measure theory, relaxation and convergence in energy},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {16},
     year = {1999},
     pages = {773-812},
     mrnumber = {1720517},
     zbl = {0943.49012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_1999__16_6_773_0}
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Sychev, M. A. A new approach to Young measure theory, relaxation and convergence in energy. Annales de l'I.H.P. Analyse non linéaire, Tome 16 (1999) pp. 773-812. http://gdmltest.u-ga.fr/item/AIHPC_1999__16_6_773_0/

[1] E. Acerbi, N. Fusco, Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal., Vol. 86, 1984, pp. 125-145. | MR 751305 | Zbl 0565.49010

[2] E. Acerbi, G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc.Var. & PDE 2 (1994), no. 3, pp. 329-371. | Zbl 0810.49014

[3] E.J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. Control and Optimization, Vol. 22, 1984, pp. 570-598. | MR 747970 | Zbl 0549.49005

[4] J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., Vol. 63, 1978, pp. 337-403. | MR 475169 | Zbl 0368.73040

[5] J.M. Ball, A version of the fundamental theorem for Young measures, in: PDE's and Continuum Models of Phase Transitions, M.Rascle, D.Serre, M.Slemrod, eds., Lecture Notes in Physics 344, Springer-Verlag (1989), pp. 207-215. | MR 1036070 | Zbl 0991.49500

[6] G. Bouchitté, I. Fonseca, J. Malý, The Effective Bulk Energy of the Relaxed Energy of Multiple Integrals Below the Growth Exponent, Proc. Royal Soc. Edinb. Sect A, Vol. 128, 1998, pp. 463-479. | MR 1632814 | Zbl 0907.49008

[7] J.M. Ball, F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., Vol. 58, 1984, pp. 225-253. | MR 759098 | Zbl 0549.46019

[8] H. Berliocchi, J.M. Lasry, Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France, Vol. 101, 1973, pp.129-184. | Numdam | MR 344980 | Zbl 0282.49041

[9] J.M. Ball, K. Zhang, Lower semicontinuity of multiple integrals and the bitting lemma, Proc. Roy. Soc. Edinburgh Sect A., Vol. 114, 1990, pp. 367-379. | MR 1055554 | Zbl 0716.49011

[10] G. Buttazzo, Lower Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Research Notes in Math. Series 207, 1989. | Zbl 0669.49005

[11] P. Celada, G.G. Dal MASO, Further remarks on the lower semicontinuity of polyconvex integrals, Ann. Inst. H. Poincaré Anal. Non. Linéaire, Vol. 11, 1994, no. 6, pp. 661-691. | Numdam | MR 1310627 | Zbl 0833.49013

[12] C. Castaing, M. Valadier, Convex analysis and measurable multifunctions, Lecture Notes in Mathematics 580, Springer-Verlag, Berlin-New York, 1977. | MR 467310 | Zbl 0346.46038

[13] A. Cellina, S. Zagatti, A version of Olech's lemma in a problem of the Calculus of Variations, SIAM J. Control and Optimization, Vol. 32, 1994, pp. 1114-1127. | MR 1280232 | Zbl 0874.49013

[14] B. Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear problems, Lecture Notes in Math. 922, Springer-Verlag, 1982. | MR 658130 | Zbl 0484.46041

[15] B. Dacorogna, Direct methods in the Calculus of Variations. Springer-Verlag, 1989. | MR 990890 | Zbl 0703.49001

[16] G. Dal Maso, C. Sbordone, Weak lower semicontinuity of polyconvex integrals: a borderline case. Math. Z., Vol. 218, 1995, no. 4, pp. 603-609. | MR 1326990 | Zbl 0822.49010

[17] R.E. Edwards, Functional Analysis, Holt, Rinehart and Winston, 1965. | MR 221256 | Zbl 0182.16101

[18] I. Ekeland, R. Temam, Convex analysis and variational problems. North-Holland, 1976. | MR 463994 | Zbl 0322.90046

[19] L.C. Evans, R.F. Gariepy, Some remarks on quasiconvexity and strong convergence, Proc. Roy. Soc. Edin. Sect. A, Vol. 106, 1987, pp. 53-61. | MR 899940 | Zbl 0628.49011

[20] L.C. Evans, Weak convergence methods for nonlinear partial differential equations. AMS, 1990. | MR 1034481 | Zbl 0698.35004

[21] N. Fusco, J. Hutchinson, A direct proof for lower semicontinuity of polyconvex functionals, Manuscripta Math., Vol. 87, 1995, no.1, pp. 35-50. Annales de l'Institut Henri Poincaré - Analyse non linéaire | MR 1329439 | Zbl 0874.49015

[22] I. Fonseca, J. Maly, Relaxation of Multiple Integrals in Sobolev Spaces Below the Growth Exponent for the Energy Density, Ann. Inst. Henri Poincaré Anal. Non. Linéaire, Vol. 14, 1997, no. 3, pp. 309-338. | Numdam | MR 1450951 | Zbl 0868.49011

[23] I. Fonseca, S. Müller, P. Pedregal, Analysis of Concentration and Oscillation Effects Generated by Gradients. To appear in SIAM J. Math. Anal. | MR 1177778 | Zbl 0920.49009

[24] M. Giaquinta, G. Modica, J. Soúcek, Remarks on lower semicontinuity of quasiconvex integrals. NODEA Nonlinear Differential Equations Appl., Vol. 2, 1995, no. 4, pp. 573-588. | MR 1356875 | Zbl 0844.49011

[25] T. Iwaniec, C. Sbordone, On the integrability of the jacobian under minimal hypotheses, Arch. Rat. Mech. Anal., Vol. 119, 1992, pp. 129-143.. | MR 1176362 | Zbl 0766.46016

[26] R. Jordan, D. Kinderlehrer, An extended variational principle, Partial differential equations and applications, M. Dekker, 1996, pp. 187-200. | MR 1371591 | Zbl 0852.49003

[27] D. Kinderlehrer, P. Pedregal, Characterization of Young measures generated by gradients, Arch. Rat. Mech. Anal., Vol. 115, 1991, pp. 329-365. | MR 1120852 | Zbl 0754.49020

[28] D. Kinderlehrer, P. Pedregal, Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal., Vol. 23, 1992, pp. 1-19. | MR 1145159 | Zbl 0757.49014

[29] D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal., Vol. 4, 1994, no 1, pp. 59-90. | MR 1274138 | Zbl 0808.46046

[30] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, Ph. D. Thesis, Technical University of Denmark, Kyngby, August 1994.

[31] K. Kuratowski, RYLL-NARDZEWSKI, A general theorem of selectors, Bull. Acad. Polon. Sci., Vol. XIII, 1966, no. 6, pp. 397-403. | MR 188994 | Zbl 0152.21403

[32] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math., Vol. 51, 1985, pp. 1-28. | MR 788671 | Zbl 0573.49010

[33] P. Marcellini, On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non Linéaire, Vol. 3, 1986, pp. 391-409. | Numdam | MR 868523 | Zbl 0609.49009

[34] J. Maly, Weak lower semicontinuity of polyconvex integrals., Proc. Roy. Soc. Edinburgh., Sect A, Vol. 123, 1993, no.4, pp. 681-691. | MR 1237608 | Zbl 0813.49017

[35] J. Maly, Lower semicontinuity of quasiconvex integrals, Manuscripta Math., Vol. 85, 1994, no.3-4, pp. 419-428. | MR 1305752 | Zbl 0862.49017

[36] C.B. Morrey, Quasi-convexity and the lower semicontinuity of multiple integrals, Pasific J. Math., Vol. 2, 1952, pp. 25-53. | MR 54865 | Zbl 0046.10803

[37] C.B. Morrey, Multiple integrals in the Calculus of Variations. Springer-Verlag, 1966. | MR 202511 | Zbl 0142.38701

[38] P. Marcellini, C. Sbordone, Semicontinuity problems in the Calculus of Variations, Nonlinear Anal., Vol. 4, 1980, pp. 241-257. | MR 563807 | Zbl 0537.49002

[39] P. Pedregal, Jensen's inequality in the calculus of variations, Differential and Integral Equations, Vol. 7, 1994, pp. 57-72. | MR 1250939 | Zbl 0810.49013

[40] P. Pedregal, Parametrized measures and variational principles. Birkhäuser, 1997. | MR 1452107 | Zbl 0879.49017

[41] Y.G. Reshetnyak, General theorems on semicontinuity and convergence with a functionals, Sibirsk. Math. Z., Vol. 8, 1967, pp. 1051-1069. | MR 220127 | Zbl 0179.20902

[42] Y.G. Reshetnyak, Space mappings with bounded distortion. Translations of Mathematical Monographs, 73, American Mathematical Society. Providence, RI, 1989. | MR 994644 | Zbl 0667.30018

[43] Y.G. Reshetnyak, A convergence theorem for functionals on additive vector-valued set functions, Sibirsk. Mat. Z., Vol. 2, 1961, pp. 115-126. | MR 130344 | Zbl 0156.14802

[44] M. Sychev, Necessary and sufficient conditions in theorems of lower semicontinuity and convergence with a functional, Russ. Acad. Sci. Sb. Math., Vol. 186, 1995, pp. 847-878. | MR 1349015 | Zbl 0835.49009

[45] M. Sychev, Young measure approach to characterization of behavior of integral functionals on weakly convergent sequences by means of their integrands, Ann. Inst. H. Poincaré Anal. Non. Linéaire, Vol. 15, 1998, no. 6. | Numdam | MR 1650962 | Zbl 0923.49009

[46] M. Sychev, Young measures as measurable functions (in preparation).

[47] M. Sychev, Characterization of homogeneous gradient Young measures in the case of arbitrary integrands. Report No. 98-216, May 98, Carnegie Mellon University.

[48] M. Sychev, New approach to Young measure theory, relaxation and convergence in energy. SISSA preprint 44/97/M, March 1997, Triest.

[49] L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Pitman Research Notes in Mathematics, Vol. 39, 1979, pp. 136-212. | MR 584398 | Zbl 0437.35004

[50] L. Tartar, The compensated compactness method applied to systems of conservations laws, in Systems of Nonlinear Partial Differential Equations, J. M. Ball ed., NATO ASI Series, Vol. CIII, Reidel, 1982. | MR 725524 | Zbl 0536.35003

[51] M. Valadier, Young measures, in Methods of Nonconvex Analysis, ed. A. Cellina, Lecture Notes in Math., Vol. 1446, 1990, pp. 152-188. | MR 1079763 | Zbl 0738.28004

[52] G.V. Vasilenko, Convergence with a functional, Sibirsk. Math. Z., Vol. 27, 1986, no. 1, pp. 26-34. | MR 847411 | Zbl 0608.49011

[53] A. Visintin, Strong convergence results related to strict convexity, Comm. Partial Differential Equations, Vol. 9, 1984, pp. 439-466. | MR 741216 | Zbl 0545.49019

[54] L.C. Young, Lectures on the Calculus of Variations and Optimal Control Theory. Saunders, 1969 (reprinted by Chelsea 1980). | MR 259704 | Zbl 0177.37801

[55] L.C. Young, Generalized curves and the existence of an attained absolute minimum in the Calculus of Variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III, Vol. 30, 1937, pp. 212-234. | JFM 63.1064.01 | Zbl 0019.21901

[56] W.P. Ziemer, Weakly differentiable functions. Springer-Verlag, New-York, 1989. | MR 1014685 | Zbl 0692.46022