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Young measure approach to characterization of behaviour of integral functionals on weakly convergent sequences by means of their integrands. (English) Zbl 0923.49009
The integral functional \(I(u)= \int_\Omega L(x,u(x),\nabla u(x))dx\) with \(u\in W^{1,p} (\Omega; \mathbb{R}^m)\), admitting also \(I(u)= +\infty\), is addressed. The sequential weak lower semicontinuity of \(I\) is related with quasiconvexity of \(L(x,u,\cdot)\). Moreover, the property that \(\{u_k\}\) converges weakly to \(u\) and \(I(u_k)\to I(u)\) implies \(\{u_k\}\) converging strongly is related with a so-called strict \(p\)-quasiconvexity. Young measure theory, presented in some detail in this paper, and equi-integrability are main tools used also to give alternative proofs of some other already known results.

49J45 Methods involving semicontinuity and convergence; relaxation
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[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. rat. mech. anal., Vol. 86, 125-145, (1984) · Zbl 0565.49010
[2] Alberti, G., A Lusin type theorem for gradients, J. funct. anal., Vol. 100, 110-118, (1991) · Zbl 0752.46025
[3] Balder, E.J., A general approach to lower semicontinuity and lower closure in optimal control theory, SIAM J. control and optimization, Vol. 22, 570-598, (1984) · Zbl 0549.49005
[4] Ball, J.M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. rat. mech. anal., Vol. 6, 337-403, (1978) · Zbl 0368.73040
[5] Ball, J.M., A version of the fundamental theorem for Young measures, (), 207-215 · Zbl 0991.49500
[6] Ball, J.M.; Murat, F., W^1, p-quasiconvexity and variational problems for multiple integrals, J. funct. anal., Vol. 58, 225-253, (1984) · Zbl 0549.46019
[7] Ball, J.M.; Zhang, K., Lower semicontinuity of multiple integrals and the biting lemma, (), 367-379 · Zbl 0716.49011
[8] Cellina, A., On minima of functionals of gradient: necessary conditions, Nonlinear analysis TMA, Vol. 20, 337-341, (1993) · Zbl 0784.49021
[9] Cellina, A., On minima of functionals of gradient: sufficient conditions, Nonlinear analysis TMA, Vol. 20, 343-347, (1993) · Zbl 0784.49022
[10] Cellina, A.; Zagatti, S., A version of Olech’s lemma in a problem of the calculus of variations, SIAM J. control and optimization, Vol. 32, 1114-1127, (1994) · Zbl 0874.49013
[11] Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear problems, () · Zbl 0676.46035
[12] Dacorogna, B., Direct methods in the calculus of variations, (1989), Springer-Verlag · Zbl 0703.49001
[13] Ekeland, I.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam
[14] Evans, L.C.; Gariepy, R.F., Some remarks on quasiconvexity and strong convergence, (), 53-61 · Zbl 0628.49011
[15] Friesecke, G., A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems, (), 437-471 · Zbl 0809.49017
[16] Iwaniec, T.; Sbordone, C., On the integrability of the Jacobian under minimal hypotheses, Arch. rat. mech. anal., Vol. 119, 129-143, (1992) · Zbl 0766.46016
[17] \scO. Kalamajska, Oral communication.
[18] Kinderlehrer, D.; Pedregal, P., Characterization of Young measures generated by gradients, Arch. rat. mech. anal., Vol. 115, 329-365, (1991) · Zbl 0754.49020
[19] Kinderlehrer, D.; Pedregal, P., Weak convergence of integrands and the Young measure representation, SIAM J. math. anal., Vol. 23, 1-19, (1992) · Zbl 0757.49014
[20] Kinderlehrer, D.; Pedregal, P., Gradient Young measures generated by sequences in Sobolev spaces, J. geom. anal., Vol. 4, No. 1, 59-90, (1994) · Zbl 0808.46046
[21] Kristensen, J., Finite functionals and Young measures generated by gradients of Sobolev functions, Mat-report no. 1994-34, (August 1994)
[22] Krasnosel’skij, M.; Rutickij, Y., Convex functions and Orlicz spaces, (1961), Noordhoff Groningen
[23] Kuratowski, K.; Ryll-Nardzewski, K, A general theorem of selectors, Bull. acad. polon. sci., Vol. XIII, No. 6, 397-403, (1966) · Zbl 0152.21403
[24] Knops, R.J.; Stuart, C.A., Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. rat. mech. anal., Vol. 86, No. 3, 233-249, (1984) · Zbl 0589.73017
[25] Marcellini, P., Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta math., Vol. 51, 1-28, (1985) · Zbl 0573.49010
[26] Maly, J., Weak lower semicontinuity of polyconvex integrals, (), 681-691, No. 4 · Zbl 0813.49017
[27] Morrey, C.B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag · Zbl 0142.38701
[28] Pedregal, P., Jensen’s inequality in the calculus of variations, Differential and integral equations, Vol. 7, 57-72, (1994) · Zbl 0810.49013
[29] Rudin, W., Functional analysis, (1985), Tata Mc Graw-Hill · Zbl 0613.26001
[30] Sychev, M., Necessary and sufficient conditions in theorems of lower semicontinuity and convergence with a functional, Russ. acad. sci. sb. math., Vol. 186, 847-878, (1995) · Zbl 0835.49009
[31] Sychev, M., Characterization of weak-strong convergence property of integral functionals by means of their integrands, (1994), Inst. Math. Siberian Division of Russ. Acad. Sci Novosibirsk, Preprint 11
[32] Sychev, M., A criterion for continuity of an integral functional on a sequence of functions, Siberian math. J., Vol. 36, No. 1, 146-156, (1995)
[33] Visintin, A., Strong convergence results related to strict convexity, Comm. partial differential equations, Vol. 9, 439-466, (1984) · Zbl 0545.49019
[34] Young, L.C., Lectures on the calculus of variations and optimal control theory, (1969), Saunders, (reprinted by Chelsea, 1980) · Zbl 0177.37801
[35] Young, L.C., 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, Vol. 30, 212-234, (1937), classe III · Zbl 0019.21901
[36] Ziemer, W.P., Weakly differentiable functions, (1989), Springer-Verlag New-York · Zbl 0177.08006
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