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Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design


Author: Pablo Pedregal
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 72-78
MSC (2000): Primary 49J45, 74P10
DOI: https://doi.org/10.1090/S1079-6762-01-00096-8
Published electronically: August 22, 2001
MathSciNet review: 1856792
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Abstract | References | Similar Articles | Additional Information

Abstract: We explicitly compute the quasiconvexification of the resulting integrand associated with the mean-square deviation of the gradient of the state with respect to a given target field, when the underlying optimal design problem in conductivity is reformulated as a purely variational problem. What is remarkable, more than the formula itself, is the fact that it can be shown to be the full quasiconvexification.


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  • 1. Bellido, J. C. and Pedregal, P., Optimal design via variational principles: the one-dimensional case, J. Math. Pures Appl. 80 (2001), 245-261. CMP 2001:08
  • 2. Bellido, J. C. and Pedregal, P., in preparation.
  • 3. Dacorogna, B., Direct methods in the Calculus of Variations, Springer, 1989. MR 90e:49001
  • 4. Grabovsky, Y., Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, 2001, to appear in Advan. Appl. Math.
  • 5. Kohn, R., The relaxation of a double-well energy, Cont. Mech. Thermodyn. 3 (1991), 193-236. MR 93d:73014
  • 6. Kohn, R. V. and Strang, G., Optimal design and relaxation of variational problems, I, II and III, CPAM 39 (1986), 113-137, 139-182 and 353-377. MR 87d:49019a; MR 87d:49019b; MR 87i:49023
  • 7. Lipton, R. and Velo, A., 2000 Optimal design of gradient fields with applications to electrostatics, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, D. Cioranescu, F. Murat, and J. L. Lions, eds., Chapman and Hall/CRC Research Notes in Mathematics.
  • 8. Murat, F., Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura ed Appl., Serie 4 112 (1977), 49-68. MR 55:11123
  • 9. Pedregal, P., Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997. MR 98e:49001
  • 10. Pedregal, P., Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32 (2000), 854-869. CMP 2001:08
  • 11. Sverák, V., Lower semicontinuity of variational integrals and compensated compactness, in S. D. Chatterji, ed., Proc. ICM, vol. 2, Birkhäuser, 1994, pp. 1153-1158. MR 97h:49021
  • 12. Tartar, L., Remarks on optimal design problems, in Calculus of Variations, Homogenization and Continuum Mechanics, G. Buttazzo, G. Bouchitté and P. Suquet, eds., World Scientific, Singapore, 1994, pp. 279-296. MR 97g:49001
  • 13. Tartar, L., 2000 An introduction to the homogenization method in optimal design, Springer Lecture Notes in Math., vol. 1740, pp. 47-156. CMP 2001:07

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Additional Information

Pablo Pedregal
Affiliation: Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
Email: ppedrega@ind-cr.uclm.es

DOI: https://doi.org/10.1090/S1079-6762-01-00096-8
Received by editor(s): March 15, 2001
Published electronically: August 22, 2001
Additional Notes: I would like to acknowledge several stimulating conversations with R. Lipton concerning the type of optimal design problems considered here and to J. C. Bellido for carrying out various initial computations. I also appreciate the criticism of several referees which led to the improvement of several aspects of this note.
Communicated by: Stuart Antman
Article copyright: © Copyright 2001 American Mathematical Society

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