Remote Access Electronic Research Announcements

Electronic Research Announcements

ISSN 1079-6762



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
Published electronically: August 22, 2001
MathSciNet review: 1856792
Full-text PDF Free Access

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.

References [Enhancements On Off] (What's this?)

  • 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. Bernard Dacorogna, Direct methods in the calculus of variations, Applied Mathematical Sciences, vol. 78, Springer-Verlag, Berlin, 1989. MR 990890
  • 4. Grabovsky, Y., Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals, 2001, to appear in Advan. Appl. Math.
  • 5. R. V. Kohn, The relaxation of a double-well energy, Contin. Mech. Thermodyn. 3 (1991), no. 3, 193–236. MR 1122017, 10.1007/BF01135336
  • 6. 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, 10.1002/cpa.3160390107
    Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. II, Comm. Pure Appl. Math. 39 (1986), no. 2, 139–182. MR 820067, 10.1002/cpa.3160390202
    Robert V. Kohn and Gilbert Strang, Optimal design and relaxation of variational problems. III, Comm. Pure Appl. Math. 39 (1986), no. 3, 353–377. MR 829845, 10.1002/cpa.3160390305
  • 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. François Murat, Contre-exemples pour divers problèmes où le contrôle intervient dans les coefficients, Ann. Mat. Pura Appl. (4) 112 (1977), 49–68. MR 0438205
  • 9. Pablo Pedregal, Parametrized measures and variational principles, Progress in Nonlinear Differential Equations and their Applications, 30, Birkhäuser Verlag, Basel, 1997. MR 1452107
  • 10. Pedregal, P., Optimal design and constrained quasiconvexity, SIAM J. Math. Anal. 32 (2000), 854-869. CMP 2001:08
  • 11. Vladimír Šverák, Lower-semicontinuity of variational integrals and compensated compactness, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 1153–1158. MR 1404015
  • 12. 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
  • 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

Similar Articles

Retrieve articles in Electronic Research Announcements of the American Mathematical Society with MSC (2000): 49J45, 74P10

Retrieve articles in all journals with MSC (2000): 49J45, 74P10

Additional Information

Pablo Pedregal
Affiliation: Departamento de Matemáticas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

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