A note on Young measures and correctors in $\Gamma$-convergence and homogenization
Author:
Pablo Pedregal
Journal:
Quart. Appl. Math. 68 (2010), 661-669
MSC (2010):
Primary 35B27, 49J45, 74Q05
DOI:
https://doi.org/10.1090/S0033-569X-2010-01184-8
Published electronically:
September 3, 2010
MathSciNet review:
2761509
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We explore a general strategy to determine the Young measure associated with sequences of pairs of conductivity coefficients and corresponding gradient fields. We also relate this issue to corrector results and investigate various examples.
References
- Grégoire Allaire, Shape optimization by the homogenization method, Applied Mathematical Sciences, vol. 146, Springer-Verlag, New York, 2002. MR 1859696
- Erik J. Balder, Lectures on Young measure theory and its applications in economics, Rend. Istit. Mat. Univ. Trieste 31 (2000), no. suppl. 1, 1–69. Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997). MR 1798830
- Marc Briane, Correctors for the homogenization of a laminate, Adv. Math. Sci. Appl. 4 (1994), no. 2, 357–379. MR 1294225
- Doina Cioranescu and Patrizia Donato, An introduction to homogenization, Oxford Lecture Series in Mathematics and its Applications, vol. 17, The Clarendon Press, Oxford University Press, New York, 1999. MR 1765047
- Gianni Dal Maso, An introduction to $\Gamma $-convergence, Progress in Nonlinear Differential Equations and their Applications, vol. 8, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1201152
- Gianni Dal Maso and Anneliese Defranceschi, Correctors for the homogenization of monotone operators, Differential Integral Equations 3 (1990), no. 6, 1151–1166. MR 1073064
- Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. I, Comm. Pure Appl. Math. 39 (1986), no. 1, 113–137. MR 820342, DOI https://doi.org/10.1002/cpa.3160390107
- François Murat and Luc Tartar, $H$-convergence, Topics in the mathematical modelling of composite materials, Progr. Nonlinear Differential Equations Appl., vol. 31, Birkhäuser Boston, Boston, MA, 1997, pp. 21–43. MR 1493039
- Alexander Pankov, $G$-convergence and homogenization of nonlinear partial differential operators, Mathematics and its Applications, vol. 422, Kluwer Academic Publishers, Dordrecht, 1997. MR 1482803
- Pablo Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications, vol. 30, Birkhäuser Verlag, Basel, 1997. MR 1452107
- Pablo Pedregal, $\Gamma $-convergence through Young measures, SIAM J. Math. Anal. 36 (2004), no. 2, 423–440. MR 2111784, DOI https://doi.org/10.1137/S0036141003425696
- Pablo Pedregal, Young measures associated with homogenization, SIAM J. Math. Anal. 37 (2006), no. 5, 1454–1464. MR 2215272, DOI https://doi.org/10.1137/040614578
- Serrano, H., Ph.D. Thesis, U. Complutense, 2007.
- Michel Valadier, Young measures, Methods of nonconvex analysis (Varenna, 1989) Lecture Notes in Math., vol. 1446, Springer, Berlin, 1990, pp. 152–188. MR 1079763, DOI https://doi.org/10.1007/BFb0084935
References
- Allaire, G. Shape optimization by the homogenization method, 2002, Springer. MR 1859696 (2002h:49001)
- Balder, E. J. (2000), Lectures on Young measure theory and its applications in economics, in: Proceedings Grado School on Measure Theory and Real Analysis, Rend. Sem. Matem. Trieste 31, suppl. 1, 1–69. MR 1798830 (2001m:49069)
- Briane, M., Correctors for the homogenization of a laminate, Adv. Math. Sci. Appl., Gakk. Tokyo, 4, n. 2 (1994), 357–379. MR 1294225 (95h:35023)
- Cioranescu, D., Donato, P., An Introduction to Homogenization, 1999, Oxford Univ. Press. MR 1765047 (2001j:35019)
- Dal Maso, G. (1993), Introduction to $\Gamma$-convergence, Boston MA, Birkhäuser. MR 1201152 (94a:49001)
- Dal Maso, G., Defranceschi, A., Correctors for the homogenization of monotone operators, Diff. Int. Equations 3 (1990), no. 6, 1151-1166. MR 1073064 (91k:35032)
- Kohn, R. V., Strang, G. (1986), Optimal design and relaxation of variational problems, I, II and III, CPAM, 39, 113-137, 139-182 and 353-377. MR 820342 (87d:49019a)
- Murat, F., Tartar, L., H-convergence, in Topics in the mathematical modelling of composite materials, Cherkaev, A., Kohn, R. V., eds., Boston, Birkhäuser, 1997, 21-44. MR 1493039
- Pankov, A., $G$-convergence and homogenization of nonlinear partial differential operators, Mathematics and its Applications, 422. Kluwer Academic Publishers, Dordrecht, 1997. MR 1482803 (99a:35017)
- Pedregal, P. (1997), Parametrized Measures and Variational Principles, Birkhäuser, Basel. MR 1452107 (98e:49001)
- Pedregal, P., $\Gamma$-convergence through Young measures, SIAM J. Math. Anal., 36, n. 2 (2004), 423–440. MR 2111784 (2005i:49016)
- Pedregal, P., Young measures associated with homogenization, SIAM J. Math. Anal., 37, n. 5 (2006), 1454-1464. MR 2215272 (2006m:49023)
- Serrano, H., Ph.D. Thesis, U. Complutense, 2007.
- Valadier M. (1990), Young measures, Methods of Nonconvex Analysis, Lect. Notes in Math., 1446, Springer, 152-188. MR 1079763 (91j:28006)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC (2010):
35B27,
49J45,
74Q05
Retrieve articles in all journals
with MSC (2010):
35B27,
49J45,
74Q05
Additional Information
Pablo Pedregal
Affiliation:
Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
Email:
pablo.pedregal@uclm.es
Received by editor(s):
February 13, 2009
Published electronically:
September 3, 2010
Additional Notes:
Supported by project MTM2007-62945 (MEC, Spain) and by project PCI08-0084-0424 from JCCM (Castilla-La Mancha).
Article copyright:
© Copyright 2010
Brown University
The copyright for this article reverts to public domain 28 years after publication.
