The purpose of this note is to discuss the relationship among Rosenthal's modulus of uniform integrability, Young measures and DiPerna-Majda measures. In particular, we give an explicit characterization of this modulus and state a criterion of the uniform integrability in terms of these measures. Further, we show applications to Fatou's lemma.
@article{119029, author = {Martin Kru\v z\'\i k}, title = {DiPerna-Majda measures and uniform integrability}, journal = {Commentationes Mathematicae Universitatis Carolinae}, volume = {39}, year = {1998}, pages = {511-523}, zbl = {0970.49012}, mrnumber = {1666786}, language = {en}, url = {http://dml.mathdoc.fr/item/119029} }
Kružík, Martin. DiPerna-Majda measures and uniform integrability. Commentationes Mathematicae Universitatis Carolinae, Tome 39 (1998) pp. 511-523. http://gdmltest.u-ga.fr/item/119029/
On weak convergence implying strong convergence in $L_1$ spaces, Bull. Austral. Math. Soc. 33 (1986), 363-368. (1986) | MR 0837481
On equivalence of strong and weak convergence in $L_1$-spaces under extreme point conditions, Israel J. Math. 75 (1991), 21-47. (1991) | MR 1147289
On weak convergence implying strong convergence under extremal conditions, J. Math. Anal. Appl. 163 (1992), 147-156. (1992) | MR 1144712 | Zbl 0768.46013
A version of the fundamental theorem for Young measures, in: PDEs and Continuum Models of Phase Transition (M. Rascle, D. Serre, M. Slemrod, eds.), Lecture Notes in Physics, Vol. 344, Springer-Verlag, 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
Martingales valued in certain subspaces of $L^1$, Israel J. Math. 37 (1980), 54-75. (1980) | MR 0599302
Continuity and compactness of measures, Adv. in Math. 37 (1980), 16-26. (1980) | MR 0585896 | Zbl 0463.28003
Méthode de compacticité et de décomposition applications: minimisation, convergence des martingales, lemme de Fatou multivoque, Ann. Mat. Pura Appl. 164 (1993), 51-75. (1993) | MR 1243948
Weak Continuity and Weak Lower Semicontinuity of Non-Linear Functionals, Lecture Notes in Math., Vol. 922, Springer-Verlag, Berlin, 1982. | MR 0658130
Sur l'intégrale de Lebesgue, Trans. Amer. Math. Soc. 16 (1915), 435-501. (1915) | MR 1501024
Oscillations and concentrations in weak solutions of the incompressible fluid equations, Math. Physics 108 (1987), 667-689. (1987) | MR 0877643 | Zbl 0626.35059
Linear Operators, Part I, Interscience, New York, 1967. | Zbl 0635.47003
General Topology, $2^{nd}$ ed., PWN, Warszawa, 1985. | Zbl 0684.54001
Weak Convergence Methods for Nonlinear Partial Differential Equations, A.M.S., Providence, 1990. | MR 1034481 | Zbl 0698.35004
Analysis of concentration and oscillation effects generated by gradients, preprint. | MR 1617712
Weak convergence of integrands and the Young measure representation, SIAM J. Math. Anal. 23 (1992), 1-19. (1992) | MR 1145159 | Zbl 0757.49014
Gradient Young measures generated by sequences in Sobolev spaces, J. Geom. Anal. 4 (1994), 59-90. (1994) | MR 1274138 | Zbl 0828.46031
Une extension du lemme de Fatou, Bull. Sci. Math. $2^{nd}$ serie 115 (1991), 211-221. (1991) | MR 1101024 | Zbl 0734.28004
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 122 (1997), 383-399. (1997) | MR 1489400
Uniform Integrability and Young measures, J. Math. Anal. Appl. 195 (1995), 428-439. (1995) | MR 1354553 | Zbl 0930.28004
Nonconcentrating generalized Young functionals, Comment. Math. Univ. Carolinae 38 (1997), 91-99. (1997) | MR 1455472
Relaxation in Optimization Theory and Variational Calculus, W. de Gruyter, Berlin, 1997. | MR 1458067
Extraction of a ``good'' sequence from a bounded sequence of integrable functions, J. Convex Anal. 2 (1994), 345-357. (1994)
Convergence of solutions to nonlinear dispersive equations, Comm. Partial Diff. Equations 7 (1982), 959-1000. (1982) | MR 0668586 | Zbl 0496.35058
Strong convergence of vector valued pramarts and submarts, Probability and Math. Stat. 5 (1985), 187-196. (1985) | MR 0816695
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., Vol. 39, San Francisco, 1979. | MR 0584398 | Zbl 0437.35004
Young measures, in: Methods of Nonconvex Analysis (A. Cellina, ed.), Lecture Notes in Math., Vol. 1446, Springer-Verlag, Berlin, 1990, 152-188. | MR 1079763 | Zbl 1067.28001
Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972. | MR 0372708 | Zbl 0253.49001
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
Lectures on the Calculus of Variations and Optimal Control Theory, W.B. Saunders, Philadelphia, 1969. | MR 0259704 | Zbl 0289.49003