# zbMATH — the first resource for mathematics

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.

##### MSC:
 49J45 Methods involving semicontinuity and convergence; relaxation
Full Text:
##### References:
 [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
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.