| Title:
|
On convergence of gradient-dependent integrands (English) |
| Author:
|
Kružík, Martin |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
52 |
| Issue:
|
6 |
| Year:
|
2007 |
| Pages:
|
529-543 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We study convergence properties of $\lbrace v(\nabla u_k)\rbrace _{k\in \mathbb{N}}$ if $v\in C(\mathbb{R}^{m\times n})$, $|v(s)|\le C(1+|s|^p)$, $1<p<+\infty $, has a finite quasiconvex envelope, $u_k\rightarrow u$ weakly in $W^{1,p} (\Omega ;\mathbb{R}^m)$ and for some $g\in C(\Omega )$ it holds that $\int _\Omega g(x)v(\nabla u_k(x))\mathrm{d}x\rightarrow \int _\Omega g(x) Qv(\nabla u(x))\mathrm{d}x$ as $k\rightarrow \infty $. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of $\lbrace \det \nabla u_k\rbrace _{k\in \mathbb{N}}$ to $\det \nabla u$ if $m=n=p$. (English) |
| Keyword:
|
bounded sequences of gradients |
| Keyword:
|
concentrations |
| Keyword:
|
oscillations |
| Keyword:
|
quasiconvexity |
| Keyword:
|
weak convergence |
| MSC:
|
35B05 |
| MSC:
|
49J45 |
| idZBL:
|
Zbl 1164.49305 |
| idMR:
|
MR2357579 |
| DOI:
|
10.1007/s10492-007-0031-4 |
| . |
| Date available:
|
2009-09-22T18:31:46Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134694 |
| . |
| Reference:
|
[1] J. J. Alibert, G. Bouchitté: Non-uniform integrability and generalized Young measures.J. Convex Anal. 4 (1997), 129–147. MR 1459885 |
| Reference:
|
[2] J. M. Ball: A version of the fundamental theorem for Young measures.In: PDEs and Continuum Models of Phase Transition. Lect. Notes Phys. 344, M. Rascle, D. Serre, M. Slemrod (eds.), Springer-Verlag, Berlin, 1989, pp. 207–215. Zbl 0991.49500, MR 1036070 |
| Reference:
|
[3] J. M. Ball, F. Murat: $W^{1,p}$-quasiconvexity and variational problems for multiple integrals.J. Funct. Anal. 58 (1984), 225–253. MR 0759098, 10.1016/0022-1236(84)90041-7 |
| Reference:
|
[4] J. M. Ball, K.-W. Zhang: Lower semicontinuity of multiple integrals and the biting lemma.Proc. R. Soc. Edinb. 114 A (1990), 367–379. MR 1055554 |
| Reference:
|
[5] J. K. Brooks, R. V. Chacon: Continuity and compactness of measures.Adv. Math. 37 (1980), 16–26. MR 0585896, 10.1016/0001-8708(80)90023-7 |
| Reference:
|
[6] B. Dacorogna: Direct Methods in the Calculus of Variations.Springer-Verlag, Berlin, 1989. Zbl 0703.49001, MR 0990890 |
| Reference:
|
[7] R. J. DiPerna, A. J. Majda: Oscillations and concentrations in weak solutions of the incompressible fluid equations.Commun. Math. Phys. 108 (1987), 667–689. MR 0877643, 10.1007/BF01214424 |
| Reference:
|
[8] N. Dunford, J. T. Schwartz: Linear Operators. Part I.Interscience, New York, 1967. |
| Reference:
|
[9] R. Engelking: General Topology. 2nd ed.PWN, Warszawa, 1976. (Polish) MR 0500779 |
| Reference:
|
[10] I. Fonseca: Lower semicontinuity of surface energies.Proc. R. Soc. Edinb. 120 A (1992), 99–115. Zbl 0757.49013, MR 1149987 |
| Reference:
|
[11] I. Fonseca, S. Müller, P. Pedregal: Analysis of concentration and oscillation effects generated by gradients.SIAM J. Math. Anal. 29 (1998), 736–756. MR 1617712, 10.1137/S0036141096306534 |
| Reference:
|
[12] J. Hogan, C. Li, A. McIntosh, K. Zhang: Global higher integrability of Jacobians on bounded domains.Ann. Inst. Henri Poincaré, Anal. Non Linéaire 17 C (2000), 193–217. MR 1753093, 10.1016/S0294-1449(00)00108-6 |
| Reference:
|
[13] A. Kałamajska, M. Kružík: Oscillations and concentrations in sequences of gradients.ESAIM Control Optim. Calc. Var (to appear). MR 2375752 |
| Reference:
|
[14] D. Kinderlehrer, P. Pedregal: Characterization of Young measures generated by gradients.Arch. Ration. Mech. Anal. 115 (1991), 329–365. MR 1120852, 10.1007/BF00375279 |
| Reference:
|
[15] D. Kinderlehrer, P. Pedregal: Weak convergence of integrands and the Young measure representation.SIAM J. Math. Anal. 23 (1992), 1–19. MR 1145159, 10.1137/0523001 |
| Reference:
|
[16] D. Kinderlehrer, P. Pedregal: Gradient Young measures generated by sequences in Sobolev spaces.J. Geom. Anal. 4 (1994), 59–90. MR 1274138, 10.1007/BF02921593 |
| Reference:
|
[17] M. Kočvara, M. Kružík, and J. V. Outrata: On the control of an evolutionary equilibrium in micromagnetics.Optimization with multivalued mappings. Springer Optim. Appl., Vol. 2, Springer-Verlag, New York, 2006, pp. 143–168. MR 2243541 |
| Reference:
|
[18] J. Kristensen: Lower semicontinuity in spaces of weakly differentiable functions.Math. Ann. 313 (1999), 653–710. Zbl 0924.49012, MR 1686943, 10.1007/s002080050277 |
| Reference:
|
[19] M. Kružík, T. Roubíček: On the measures of DiPerna and Majda.Math. Bohem. 122 (1997), 383–399. MR 1489400 |
| Reference:
|
[20] M. Kružík, T. Roubíček: Optimization problems with concentration and oscillation effects: Relaxation theory and numerical approximation.Numer. Funct. Anal. Optimization 20 (1999), 511–530. MR 1704958, 10.1080/01630569908816908 |
| Reference:
|
[21] C. B. Morrey: Multiple Integrals in the Calculus of Variations.Springer-Verlag, Berlin, 1966. Zbl 0142.38701, MR 0202511 |
| Reference:
|
[22] S. Müller: Higher integrability of determinants and weak convergence in $L^1$.J. Reine Angew. Math. 412 (1990), 20–34. MR 1078998 |
| Reference:
|
[23] S. Müller: Variational models for microstructure and phase transisions.Lect. Notes Math. 1713, Springer-Verlag, Berlin, 1999, pp. 85–210. MR 1731640 |
| Reference:
|
[24] P. Pedregal: Parametrized Measures and Variational Principles.Birkäuser-Verlag, Basel, 1997. Zbl 0879.49017, MR 1452107 |
| Reference:
|
[25] T. Roubíček: Relaxation in Optimization Theory and Variational Calculus.W. de Gruyter, Berlin, 1997. MR 1458067 |
| Reference:
|
[26] M. E. Schonbek: Convergence of solutions to nonlinear dispersive equations.Comm. Partial Differ. Equations 7 (1982), 959–1000. Zbl 0496.35058, MR 0668586, 10.1080/03605308208820242 |
| Reference:
|
[27] L. Tartar: Compensated compactness and applications to partial differential equations.In: Nonlinear Analysis and Mechanics. Heriot-Watt Symposium IV. Res. Notes Math. 39, R. J. Knops (ed.), , San Francisco, 1979. Zbl 0437.35004, MR 0584398 |
| Reference:
|
[28] L. Tartar: Mathematical tools for studying oscillations and concentrations: From Young measures to $H$-measures and their variants.In: Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Proceedings of the conference on multiscale problems in science and technology, held in Dubrovnik, Croatia, September 3–9, 2000, N. Antonič et al. (eds.), Springer-Verlag, Berlin, 2002, pp. 1–84. Zbl 1015.35001, MR 1998790 |
| Reference:
|
[29] M. Valadier: Young measures.In: Methods of Nonconvex Analysis. Lect. Notes Math. 1446, A. Cellina (ed.), Springer-Verlag, Berlin, 1990, pp. 152–188. Zbl 0742.49010, MR 1079763 |
| Reference:
|
[30] J. Warga: Optimal Control of Differential and Functional Equations.Academic Press, New York, 1972. Zbl 0253.49001, MR 0372708 |
| Reference:
|
[31] L. C. Young: Generalized curves and the existence of an attained absolute minimum in the calculus of variations.C. R. Soc. Sci. Lett. Varsovie, Classe III 30 (1937), 212–234. Zbl 0019.21901 |
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