Advertisement

What Does Rank-One Convexity Have to Do with Viscosity Solutions?

  • Pablo PedregalEmail author
Chapter
  • 509 Downloads
Part of the Springer INdAM Series book series (SINDAMS, volume 27)

Abstract

Relying on Hilbert’s classical theorem for non-negative polynomials as a main tool, we show that rank-one convex functions for 2 × 2-matrices admit a decomposition as a sum of a multiple of the determinant and a viscosity solution of a certain equation.

Notes

Acknowledgements

Partially supported by MINECO/FEDER grant MTM2013-47053-P, by PEII-2014-010-P of the Conserjería de Cultura (JCCM), and by grant GI20152919 of UCLM.

References

  1. 1.
    Antman, S.S.:, Nonlinear Problems of Elasticity, 2nd edn. Applied Mathematical Sciences, vol 107. Springer, New York (2005)Google Scholar
  2. 2.
    Artin, E.: Über die Zerlegung definiter Funktionen in Quadrate. Abh. Math. Sem. Univ. Hamburg 5(1), 100–115 (1927)Google Scholar
  3. 3.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63, 337–403 (1977)Google Scholar
  4. 4.
    Bandeira, L., Pedregal, P.: The role of non-negative polynomials for rank-one convexity and quasi convexity. J. Ellipic Parab. Equ. 2(2), 27–36 (2016)Google Scholar
  5. 5.
    Ciarlet, P.G.: Mathematical Elasticity. Vol. I. Three-Dimensional Elasticity. Studies in Mathematics and Its Applications, vol. 20. North-Holland, Amsterdam (1988)Google Scholar
  6. 6.
    Crandall, M.G., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)Google Scholar
  7. 7.
    Dacorogna, B.: Direct Methods in the Calculus of Variations, 2nd edn. Springer, New York (2008)Google Scholar
  8. 8.
    Hilbert, D.: Über die Darstellung Definiter Formen als Summe von Formenquadraten. Mathematische Annalen 32, 342–250 (1888)Google Scholar
  9. 9.
    Hilbert, D.: Mathematische Probleme, Lecture, Second Internat. Congr. Math. (Paris, 1900), Nachr. Ges. Wiss. Göttingen Math. Phys. KL., 253–297 (1900); English transl., Bull. Am. Math. Soc. 8, 437–479 (1902); Bull. (New Series) Am. Math. Soc. 37, 407–436 (2000)Google Scholar
  10. 10.
    Laserre, J.B.: Moments, Positive Polynomials and Their Applications. Imperial College Press, London (2010)Google Scholar
  11. 11.
    Marcellini, P.: Quasi convex quadratic forms in two dimensions. Appl. Math. Optim. 11(2), 183–189 (1984)Google Scholar
  12. 12.
    Morrey, C.B.: Quasiconvexity and the lower semicontinuity of multiple integrals. Pac. J. Math. 2, 25–53 (1952)Google Scholar
  13. 13.
    Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, New York (1966)Google Scholar
  14. 14.
    Nie, J.: Discriminants and nonnegative polynomials. J. Symb. Comp. 47, 167–191 (2012)Google Scholar
  15. 15.
    Pedregal, P., Šverák, V.: A note on quasiconvexity and rank-one convexity for 2x2 matrices. J. Convex Anal. 5(1), 107–117 (1998)Google Scholar
  16. 16.
    Šverák, V.: Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. 120 A, 293–300 (1992)Google Scholar
  17. 17.
    Tonelli, L.: Fondamenti di calcolo delle variazioni. Zanichelli, Bologna (1921)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.INEI, U. Castilla-La ManchaCiudad RealSpain

Personalised recommendations