Vector variational problems and applications to optimal design

ESAIM - Control, Optimisation and Calculus of Variations

Volumn 11, Issue 3, 2005, Pages 357-381

  Pedregal, Pablo ^a  

^a UNIVERSITY OF CASTILLA LA MANCHA   (Spain)

Author keywords

Effective; Gradient Young measures; Homogenized or relaxed integrand; Laminates

Indexed keywords

APPROXIMATION THEORY; COMPUTER SIMULATION; CONDUCTIVE MATERIALS; OPTIMAL CONTROL SYSTEMS; PROBLEM SOLVING; VECTORS;

EFFECTIVE; GRADIENT YOUNG MEASURES; HOMOGENIZED OR RELAXED INTEGRAND; LAMINATES;

VARIATIONAL TECHNIQUES;

EID: 30844437297     PISSN: 12928119     EISSN: 12623377     Source Type: Journal    
DOI: 10.1051/cocv:2005010     Document Type: Article

References (40)

1. G. Allaire, Shape optimization by the homogenization method. Springer (2002).
2. S. Antman, Nonlinear Problems of Elasticity. Springer (1995).
3. E. Aranda and P. Pedregal, Constrained envelope for a general class of design problems. DCDS-A, Supplement Volume 2003(2002) 30-41.
4. E.J. Balder, Lectures on Young Measures. Cahiers de Mathématiques de la Décision No. 9517, CEREMADE, Universite Paris 9 (1995).
5. J.M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63 (1977) 337-403.
6. J.M. Ball, A version of the fundamental theorem for Young measures, PDE’s and continuum models of phase transitions, M. Rascle, D. Serre and M. Slemrod Eds. Springer. Lect. Notes Phys. 344 (1989) 207-215.
7. J.M. Ball, Some open problems in elasticity, in Geometry, Mechanics and Dynamics, P. Newton, P. Holmes, A. Weinstein Eds.Springer (2002) 3-59.
8. J.M. Ball and R.D. James, Finephase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100 (1987) 13-52.
9. J.M. Ball and F. Murat, Remarks on Chacon’sbiting lemma. Proc. AMS 107 (1989) 655-663.
10. K. Battacharya and G. Dolzmann, Relaxation of some multi-well problems. Proc. Roy. Soc. Edinb. 131A (2001) 279-320.
11. J.C. Bellido, Explicit computation of the relaxed density coming from a three-dimensional optimal design problem. Non-Lin. Anal. TMA 52 (2002) 1709-1726.
12. J.C. Bellido and P. Pedregal, Explicit quasiconvexification of some cost functionals depending on derivatives of the state in optimal design. Disc. Cont. Dyn. Syst. A 8 (2002) 967-982.
13. M.P. Bendsoe, Optimization of structural topology, shape and material. Springer (1995).
14. M. Bousselsal and M. Chipot, Relaxation of some functionals of the calculus of variations. Arch. Math. 65 (1995) 316-326.
15. M. Bousselsal and R. Le Dret, Remarks on the quasiconvex envelope of some functions depending on quadratic forms. Boll. Union. Mat. Ital. Sez. B 5 (2002) 469-486.
16. M. Bousselsal and R. Le Dret, Relaxation of functionals involving homogeneous functions and invariance of envelopes. Chinese Ann. Math. Ser. B 23 (2002) 37-52.
17. L. Carbone and R. De Arcangelis, Unbounded functionals in the Calculus of Variations, Representation, Relaxation and Homogenization, Chapman and Hall. CRC, Monographs and Surveys in Pure and Applied Mathematics. Boca Raton, Florida125 (2002)
18. P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-dimensional Elasticity. North-Holland, Amsterdam (1987).
19. B. Dacorogna, Direct methods in the Calculus of Variations. Springer (1989).
20. G. Dolzmann, B. Kirchheim, S. Muller and V. Sverak, The two-well problem in three dimensions. Calc. Var. 10 (2000) 21-40.
21. A. Donoso and P. Pedregal, Optimal design of 2-d conducting graded materials by minimizing quadratic functionals in the field. Struct. Opt. (in press) (2004).
22. A. Donoso, Optimal design modelled by Poisson’s equation in the presence of gradients in the objective. Ph.D. Thesis, Univ. Castilla-La Mancha (2004).
23. A. Donoso, Numerical simulations in 3-d heat conduction: minimizing the quadratic mean temperature gradient (2004), submitted.
24. D. Faraco, Beltrami operators and microstructure. Ph.D. Thesis, University of Helsinki (2002).
25. I. Fonseca, D. Kinderlehrer and P. Pedregal, Energy functionals depending on elastic strain and chemical composition. Calc. Var. 2 (1994) 283-313.
26. Y. Grabovsky, Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math 27 (2001) 683-704.
27. D. Kinderlehrer and P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4 (1994)59-90.
28. R. Kohn, The relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991) 193-236.
29. R.V. Kohn and G. Strang, Optimal design and relaxation of variational problems, I, II and III. CPAM 39 (1986) 113-137, 139-182 and 353-377.
30. R. Lipton and A. Velo, Optimal design of gradient fields with applications to electrostatics, in Nonlinear Partial Differential Equations Appl., College de France Seminar, D. Cioranescu, F. Murat and J.L Lions Eds. Chapman and Hall/CRCResearch Notes in Mathematics (2000).
31. Ch.B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952) 25-53.
32. Ch.B. Morrey, Multiple Integrals in the Calculus of Variations. Berlin, Springer (1966).
33. P. Pedregal, Parametrized Measures and Variational Principles. Birkhauser, Basel (1997).
34. P. Pedregal, Variational methods in nonlinear elasticity. SIAM, Philadelphia (2000).
35. P. Pedregal, Constrained quasiconvexification of the square of the gradient of the state in optimal design. QAM 62 (2004)459-470.
36. P. Pedregal, Optimal design in 2-d conductivity for quadratic functionals in the field, in Proc. NATO Advan. Meeting Non-lin. Homog., Warsaw, Poland, Kluwer (2004) 229-246.
37. P. Pedregal, Optimal design in two-dimensional conductivity for a general cost depending on the field. Arch. Rat. Mech. Anal. (2004) (in press).
38. Y. Reshetnyak, General theorems on semicontinuity and on convergence with a functional. Sibir. Math. 8 (1967) 801-816.
39. 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 (1994) 279-296.
40. L. Tartar, An introduction to the homogenization method in optimal design, Springer. Lect. Notes Math. 1740 (2000) 47-156.