Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands
Annales de l'I.H.P. Analyse non linéaire, Volume 15 (1998) no. 6, p. 755-782
@article{AIHPC_1998__15_6_755_0,
     author = {Sychev, M. A.},
     title = {Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {15},
     number = {6},
     year = {1998},
     pages = {755-782},
     zbl = {0923.49009},
     mrnumber = {1650962},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1998__15_6_755_0}
}
Sychev, M. Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands. Annales de l'I.H.P. Analyse non linéaire, Volume 15 (1998) no. 6, pp. 755-782. http://www.numdam.org/item/AIHPC_1998__15_6_755_0/

[1] E. Acerbi and 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] G. Alberti, A Lusin type theorem for gradients, J. Funct. Anal., Vol. 100, 1991, pp. 110-118. | MR 1124295 | Zbl 0752.46025

[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. 6, 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] J.M. Ball and 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

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

[8] A. Cellina, On minima of functionals of gradient: necessary conditions, Nonlinear Analysis TMA, Vol. 20, 1993, pp. 337-341. | MR 1206422 | Zbl 0784.49021

[9] A. Cellina, On minima of functionals of gradient: sufficient conditions, Nonlinear Analysis TMA, Vol. 20, 1993, pp. 343-347. | MR 1206423 | Zbl 0784.49022

[10] A. Cellina and 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

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

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

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

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

[15] G. Friesecke, A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, Proc. Roy. Soc. Edinburgh. Sect. A, Vol. 124, 1994, pp. 437-471. | MR 1286914 | Zbl 0809.49017

[16] T. Iwaniec and 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

[17] O. Kalamajska, Oral communication.

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

[19] D. Kinderlehrer and 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

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

[21] J. Kristensen, Finite functionals and Young measures generated by gradients of Sobolev functions, MAT-REPORT No. 1994-34, August 1994.

[22] M. Krasnoselskij and Y. Rutickij, Convex functions and Orlicz spaces, Groningen, Noordhoff, 1961. | Zbl 0095.09103

[23] K. Kuratowski K. and Ryll-Nardzewski, A general theorem of selectors, Bull. Acad. Polon. Sci., Vol. XIII, No. 6, 1966, pp. 397-403. | MR 188994 | Zbl 0152.21403

[24] R.J. Knops and C.A. Stuart, Quasiconvexity and Uniqueness of Equilibrium Solutions in Nonlinear Elasticity, Arch. Rat. Mech. Anal., Vol. 86, No. 3. 1984, pp. 233-249. | MR 751508 | Zbl 0589.73017

[25] 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

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

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

[28] 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

[29] W. Rudin, Functional Analysis, Tata Mc Graw-Hill, 1985. | MR 365062 | Zbl 0253.46001

[30] 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

[3 1 ] M. Sychev, Characterization of weak-strong convergence property of integral functionals by means of their integrands, Preprint 11, 1994, Inst. Math. Siberian Division of Russ. Acad. Sci., Novosibirsk.

[32] M. Sychev, A criterion for continuity of an integral functional on a sequence of functions, Siberian Math. J., Vol. 36, No. 1, 1995, pp. 146-156. | MR 1335221 | Zbl 0856.49009

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

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

[35] 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

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