The Young measures, used widely for relaxation of various optimization problems, can be naturally understood as certain functionals on suitable space of integrands, which allows readily various generalizations. The paper is focused on such functionals which can be attained by sequences whose ``energy'' (=$p$th power) does not concentrate in the sense that it is relatively weakly compact in $L^1(\Omega )$. Straightforward applications to coercive optimization problems are briefly outlined.
@article{118904,
author = {Tom\'a\v s Roub\'\i \v cek},
title = {Nonconcentrating generalized Young functionals},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {38},
year = {1997},
pages = {91-99},
zbl = {0888.49027},
mrnumber = {1455472},
language = {en},
url = {http://dml.mathdoc.fr/item/118904}
}
Roubíček, Tomáš. Nonconcentrating generalized Young functionals. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 91-99. http://gdmltest.u-ga.fr/item/118904/
Semicontinuity problems in the calculus of variations, Archive Rat. Mech. Anal. 86 (1984), 125-145. (1984) | MR 0751305 | Zbl 0565.49010
A version of the fundamental theorem for Young measures, in: PDEs and Continuum Models of Phase Transition (Eds. M. Rascle, D. Serre, M. Slemrod), Lecture Notes in Physics 344, Springer, Berlin, 1989, pp.207-215. | MR 1036070 | Zbl 0991.49500
Remarks on Chacon's biting lemma, Proc. Amer. Math. Soc. 107 (1989), 655-663. (1989) | MR 0984807 | Zbl 0678.46023
Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973), 129-184. (1973) | MR 0344980 | Zbl 0282.49041
Continuity and compactness of measures, Adv. in Math. 37 (1980), 16-26. (1980) | MR 0585896 | Zbl 0463.28003
Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, Pitman Res. Notes in Math. 207, Longmann, Harlow, 1989. | MR 1020296 | Zbl 0669.49005
Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Physics 108 (1987), 667-689. (1987) | MR 0877643 | Zbl 0626.35059
Linear operators on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323-392. (1940) | MR 0002020
Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. 23 (1992), 1-19. (1992) | MR 1145159 | Zbl 0757.49014
Lower semicontinuity of variational integrals, Ph.D. Thesis, Math. Inst., Tech. Univ. of Denmark, Lungby, 1994.
Explicit characterization of $L^p$-Young measures, J. Math. Anal. Appl. 198 (1996), 830-843. (1996) | MR 1377827
On the measures of DiPerna and Majda, Mathematica Bohemica, in print.
Convex compactifications and special extensions of optimization problems, Nonlinear Analysis, Th., Methods, Appl. 16 (1991), 1117-1126. (1991) | MR 1111622
Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin, 1996, in print. | MR 1458067
Theory of convex local compactifications with applications to Lebesgue spaces, Nonlinear Analysis, Th., Methods, Appl. 25 (1995), 607-628. (1995) | MR 1338806
Compensated compactness and applications to partial differential equations, in: Nonlinear Analysis and Mechanics (R.J. Knops, ed.), Heriott-Watt Symposium IV, Pitman Res. Notes in Math. 39, San Francisco, 1979. | MR 0584398 | Zbl 0437.35004
Young measures, in: Methods of Nonconvex Analysis (A. Cellina, ed.), Lecture Notes in Math. 1446, Springer, Berlin, 1990, pp.152-188. | MR 1079763 | Zbl 1067.28001
Variational problems with unbounded controls, SIAM J. Control 3 (1965), 424-438. (1965) | MR 0194951 | Zbl 0201.47803
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 30 (1937), 212-234. (1937) | Zbl 0019.21901
Generalized surfaces in the calculus of variations, Ann. Math. 43 (1942), part I: 84-103, part II: 530-544. (1942) | MR 0006023 | Zbl 0063.09081