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: 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.
BellidoPedregalA Bellido, J. C. and Pedregal, P., Optimal design via variational principles: the one-dimensional case, J. Math. Pures Appl. 80 (2001), 245–261.
BellidoPedregalE Bellido, J. C. and Pedregal, P., in preparation.
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
Grabovsky Grabovsky, Y., Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, 2001, to appear in Advan. Appl. Math.
- R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn. 3 (1991), no. 3, 193–236. MR 1122017, DOI https://doi.org/10.1007/BF01135336
KohnStrang 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. ; ;
LiptonVeloA 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.
- François Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4) 112 (1977), 49–68. MR 438205, DOI https://doi.org/10.1007/BF02413475
- Pablo Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications, vol. 30, Birkhäuser Verlag, Basel, 1997. MR 1452107
PedregalN Pedregal, P., Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32 (2000), 854–869.
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- Guy Bouchitté, Giuseppe Buttazzo, and Pierre Suquet (eds.), Calculus of variations, homogenization and continuum mechanics, Series on Advances in Mathematics for Applied Sciences, vol. 18, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1428687
TartarM Tartar, L., 2000 An introduction to the homogenization method in optimal design, Springer Lecture Notes in Math., vol. 1740, pp. 47–156.
BellidoPedregalA Bellido, J. C. and Pedregal, P., Optimal design via variational principles: the one-dimensional case, J. Math. Pures Appl. 80 (2001), 245–261.
BellidoPedregalE Bellido, J. C. and Pedregal, P., in preparation.
DacorognaH Dacorogna, B., Direct methods in the Calculus of Variations, Springer, 1989.
Grabovsky Grabovsky, Y., Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, 2001, to appear in Advan. Appl. Math.
KohnA Kohn, R., The relaxation of a double-well energy, Cont. Mech. Thermodyn. 3 (1991), 193–236.
KohnStrang 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. ; ;
LiptonVeloA 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.
MuratE 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.
PedregalI Pedregal, P., Parametrized Measures and Variational Principles, Birkhäuser, Basel, 1997.
PedregalN Pedregal, P., Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32 (2000), 854–869.
SverakI Šverák, V., Lower semicontinuity of variational integrals and compensated compactness, in S. D. Chatterji, ed., Proc. ICM, vol. 2, Birkhäuser, 1994, pp. 1153–1158.
TartarK 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.
TartarM Tartar, L., 2000 An introduction to the homogenization method in optimal design, Springer Lecture Notes in Math., vol. 1740, pp. 47–156.
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Pablo Pedregal
Affiliation:
Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
Email:
ppedrega@ind-cr.uclm.es
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
