Constrained quasiconvexification of the square of the gradient of the state in optimal design
Author:
Pablo Pedregal
Journal:
Quart. Appl. Math. 62 (2004), 459-470
MSC:
Primary 49J45; Secondary 49Q20, 74P10
DOI:
https://doi.org/10.1090/qam/2086039
MathSciNet review:
MR2086039
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Abstract: We explicitly compute the constrained quasiconvexification of the integrand associated with the square of the gradient of the state in a typical optimal design problem in which a volume constraint is enforced.
E. Aranda, A. Donoso, and P. Pedregal, in preparation
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J. C. Bellido, Ph.D. Thesis, Universidad de Sevilla
- José C. Bellido and Pablo Pedregal, Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing, Discrete Contin. Dyn. Syst. 8 (2002), no. 4, 967–982. MR 1920655, DOI https://doi.org/10.3934/dcds.2002.8.967
- Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
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R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics, in Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, D. Cioranescu, F. Murat, and J. L. Lions, eds., Chapman and Hall/CRC Research Notes in Mathematics (2000)
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E. Aranda, A. Donoso, and P. Pedregal, in preparation
K. Astala and D. Faraco, Quasiregular mappings and Young measures, preprint, U. of Jyvaskyla, 2001
J. C. Bellido, Ph.D. Thesis, Universidad de Sevilla
J. C. Bellido and P. Pedregal, Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal design, Disc. Cont. Dyn. Syst.-A, 8, 967–982 (2002)
B. Dacorogna, Direct methods in the Calculus of Variations, Springer, 1989
I. Fonseca, D. Kinderlehrer, and P. Pedregal, Energy functionals depending on elastic strain and chemical composition, Calc. Var. 2, 283–313 (1994)
Y. Grabovsky, Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, Advan. Appl. Math., 27, 683–704 (2001)
R. V. Kohn and G. Strang, Optimal design and relaxation of variational problems, I, II and III, CPAM 39, 113–137, 139–182 and 353–377 (1986)
R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics, in Nonlinear Partial Differential Equations and Their Applications, College de France Seminar, D. Cioranescu, F. Murat, and J. L. Lions, eds., Chapman and Hall/CRC Research Notes in Mathematics (2000)
F. Murat, Contre-exemples pur divers problémes ou le contrôle intervient dans les coefficients, Ann. Mat. Pura ed Appl., Serie 4, 112, 49–68 (1977)
F. Murat and L. Tartar, Calcul des variations et homogénéisation, in Les méthodes de l’homogénéisation: théorie et applications en physique. Dir. des études et recherches de l’EDF, Eyrolles, Paris, 319–370 (1985)
F. Murat and L. Tartar, H convergence, in Topics in the mathematical modelling of composite materials, A. Cherkaev, R. V. Kohn, eds., Birkhäuser, Boston, 21–44 (1997)
F. Murat and L. Tartar, On the control of coefficients in partial differential equations, in Topics in the mathematical modelling of composite materials, A. Cherkaev, R. V. Kohn, eds., Birkhäuser, Boston, 1–8 (1997)
P. Pedregal, Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32, 854–869 (2000)
P. Pedregal, Fully explicit quasiconvexification of the mean-square deviation of the gradient of the state in optimal design, ERA-AMS 7, 72–78 (2001)
V. Sverak, New examples of quasiconvex functions, Arch. Rat. Mech. Anal. 119, 293–300 (1992)
V. Sverak, Lower semicontinuity of variational integrals and compensated compactness, in Proc. ICM, S. D. Chatterji, ed., vol. 2, Birkhäuser, 1153–1158 (1994)
L. Tartar, Estimations fines des coefficients homogénéises, Ennio De Giorgi Colloquium, Res. Notes in Math., P. Krée, ed., 125, Pitman, London, 168–187 (1985)
L. Tartar, On mathematical tools for studying partial differential equations of continuum physics: H-measures and Young measures, in Developments in Partial Differential Equations and Applications to Mathematical Physics (Buttazzo, Galdi, Zanghirati, eds.), Plenum, New York (1992)
L. Tartar, Remarks on optimal design problems, in Calculus of Variations, Homogenization and Continuum Mechanics, G. Buttazzo, G. Bouchitte and P. Suquet, eds., World Scientific, Singapore, 279–296 (1994)
L. Tartar, An introduction to the homogenization method in optimal design, Springer Lecture Notes in Math, 1740, 47–156 (2000)
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© Copyright 2004
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