Skip to main content
SpringerLink
Log in
Book cover

Optimal Design through the Sub-Relaxation Method pp 23–36Cite as

Our Approach

  • Chapter
  • First Online:

  • 444 Accesses

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 11))

Abstract

To introduce our analytical strategy, let us focus on the particular situation described in Sect. 1.1, but changed in a way to avoid any distraction from our main objective:

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Aranda, E., Pedregal, P.: Constrained envelope for a general class of design problems. In: Dynamical Systems and Differential Equations (Wilmington, NC, 2002). Discrete and Continuous Dynamical Systems, Supplement, pp. 30–41 (2003)

    Google Scholar 

  2. Ball, J.M., James, R.D.: Fine phase mixtures as minimizers of energy. Arch. Ration. Mech. Anal. 100, 13–52 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, J.M.: A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Physics, vol. 344, pp. 207–215. Springer, Berlin (1989)

    Google Scholar 

  4. Bellido, J.C., Pedregal, P.: Optimal design via variational principles: the one-dimensional case. J. Math. Pures Appl. 80, 245–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bellido, J.C., Pedregal, P.: Optimal design via variational principles: the three-dimensional case. J. Math. Anal. Appl. 287, 157–176 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Briane, M., Casado-Díaz, J., Murat, F.: The div-curl lemma “trente ans aprés”: an extension and an application to the G-convergence of unbounded monotone operators. J. Math. Pures Appl. 91, 476–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Castaing, Ch., Raynaud de Fitte, P., Valadier, M.: Young measures on topological spaces. In: With Applications in Control Theory and Probability Theory. Mathematics and Its Applications, vol. 571. Kluwer Academic Publishers, Dordrecht (2004)

    Google Scholar 

  8. DiPerna, R.J.: Nonuniform Structures in Solutions to Conservative Systems. Analyse mathématique et applications, pp. 139–149. Gauthier-Villars, Montrouge (1988)

    Google Scholar 

  9. Fonseca, I., Müller, S.: A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal. 30, 1355–1390 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal. 4, 59–90 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Maestre, F., Pedregal, P.: Quasiconvexification in 3-D for a variational reformulation of an optimal design problem in conductivity. Nonlinear Anal. 64, 1962–1976 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Málek, J., Nečas, J., Rokyta, M., Růžička, M.: Weak and Measure-Valued Solutions to Evolutionary PDEs. Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)

    Google Scholar 

  13. Müller, S.: Microstructures, phase transitions and geometry. In: European Congress of Mathematics, Vol. II (Budapest, 1996). Progress in Mathematics, vol. 169, pp. 92–115. Birkhäuser, Basel (1998)

    Google Scholar 

  14. Murat, F.: A survey on compensated compactness. In: Cesari, L. (ed.) Contributions to the Modern Calculus of Variations, Pitman, pp. 145–183 (1987)

    Google Scholar 

  15. Pedregal, P.: Parametrized Measures and Variational Principles. Progress in Nonlinear Differential Equations and their Applications, vol. 30. Birkhäuser Verlag, Basel (1997)

    Google Scholar 

  16. Pedregal, P.: Optimization, relaxation and Young measures. Bull. Am. Math. Soc. (N.S.) 36, 27–58 (1999)

    Google Scholar 

  17. Pedregal, P.: Optimal design and constrained quasiconvexity. SIAM J. Math. Anal. 32, 854–869 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pedregal, P.: Constrained quasiconvexification of the square of the gradient of the state in optimal design. Quart. Appl. Math. 62, 459–470 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Pedregal, P.: Vector variational problems and applications to optimal design. ESAIM Control Optim. Calc. Var. 11, 357–381 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pedregal, P.: Multi-scale Young measures. Trans. Am. Math. Soc. 358, 591–602 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pedregal, P.: Gradient Young measures and applications to optimal design. In: The Interaction of Analysis and Geometry. Contemporary Mathematics, vol. 424, pp. 187–199 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Pedregal, P.: Div-curl Young measures and optimal design in any dimension. Rev. Mat. Complut. 20, 239–255 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Roubicek, T.: Relaxation in Optimization Theory and Variational Calculus. de Gruyter Series in Nonlinear Analysis and Applications, vol. 4. Walter de Gruyter Co., Berlin (1997)

    Google Scholar 

  24. Sychev, M.A.: A new approach to Young measure theory, relaxation and convergence in energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 773–812 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tartar, L.: Compensated compactness and applications to partial differential equations. In: Knops, R. (ed.) Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. IV, Pitman, Research Notes in Mathematics, vol. 39, pp. 136–212 (1979)

    MathSciNet  Google Scholar 

  26. Tartar, L.: Beyond Young measures. Microstructure and phase transitions in solids (Udine, 1994). Meccanica 30, 505–526 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tartar, L.: An introduction to the homogenization method in optimal design. In: Optimal Shape Design (Tróia, 1998). Lecture Notes in Mathematics, vol. 1740, pp. 47–156. Springer, Berlin (2000)

    Google Scholar 

  28. Tartar, L.: Mathematical Tools for Studying Oscillations and Concentrations: From Young Measures to H-measures and Their Variants. Multiscale Problems in Science and Technology (Dubrovnik, 2000), pp. 1–84. Springer, Berlin (2002)

    Google Scholar 

  29. Valadier, M.: Young measures. In: Methods of Nonconvex Analysis (Varenna, 1989). Lecture Notes in Mathematics, vol. 1446, pp. 152–188. Springer, Berlin (1990)

    Google Scholar 

  30. Young, L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III 30, 212–234 (1937)

    MATH  Google Scholar 

  31. Young, L.C.: Generalized surfaces in the calculus of variations, I and II. Ann. Math. 43, 84–103, 530–544 (1942)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad de Castilla-La Mancha, Ciudad Real, Spain

    Pablo Pedregal

Authors
  1. Pablo Pedregal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Pedregal, P. (2016). Our Approach. In: Optimal Design through the Sub-Relaxation Method. SEMA SIMAI Springer Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-41159-0_2

Download citation

Publish with us

Policies and ethics

Search

Navigation