Numerische Mathematik

, Volume 58, Issue 1, pp 135–144 | Cite as

A numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity

  • Pablo V. Negrón Marrero


In this paper we describe and analyse a numerical method that detects singular minimizers and avoids the Lavrentiev phenomenon for three dimensional problems in nonlinear elasticity. This method extends to three dimensions the corresponding one dimensional method of Ball and Knowles.

Subject classification

AMS(MOS): 65N30 CR: G1.8 


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  1. 1.
    Adams, R.A.: Sobolev Spaces, Ed. New York: Academic Press 1975Google Scholar
  2. 2.
    Ball, J.M.: On the calculus of variations and sequentially weakly continuous maps. In: Ordinary and partial differential equations (Everitt, W.N., Sleeman, B.D. (eds.)), Lecture Notes in Mathematics, Vol. 564, pp. 13–25, Berlin New York: Springer 1976Google Scholar
  3. 3.
    Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63, 337–403 (1977)Google Scholar
  4. 4.
    Ball, J.M.: Constitutive inequalities and existence theorems in nonlinear elastostatics, In: Nonlinear analysis and mechanics: Heriot-Watt Symposium Vol. 1 (Knops, R.J. (Ed.)), London: Pitman 1977Google Scholar
  5. 5.
    Ball, J.M.: Global invertibility of Sobolev functions and the interpenetration of matter. Proc. Royal Soc. Edinburgh A88, 315–328 (1981)Google Scholar
  6. 6.
    Ball, J.M.: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. Royal Soc. London A306, 557–611 (1982)Google Scholar
  7. 7.
    Ball, J.M.: Singular minimizers and their significance in elasticity, directions in partial differential equations, (Ed.). New York: Academic Press 1987Google Scholar
  8. 8.
    Ball, J.M., Knowles, G.: A numerical method for detecting singular minimizers. Numer. Math.51, 181–197 (1987)Google Scholar
  9. 9.
    Ball, J.M., Mizel, V.J.: Singular minimizers for regular one dimensional problems in the calculus of variations. Bull. Am. Math. Soc. New Ser.11, 143–146 (1984)Google Scholar
  10. 10.
    Ball, J.M., Mizel, V.J.: One dimensional variational problems whose minimizers do not satisfy the Euler-Langrange equation. Arch. Rat. Mech. Anal.90, 325–388 (1985)Google Scholar
  11. 11.
    Ball, J.M., Currie, J.C., Olver, P.J.: Null lagrangians, weak continuity, and variational problems of arbitrary order. J. Funct. Anal.41, 135–174 (1981)Google Scholar
  12. 12.
    Cesari, L.: Optimization — theory and applications, (Ed.). Berlin Heidelberg New York: Springer 1983Google Scholar
  13. 13.
    Ciarlet, P.G., Nečas, J.: Injectivity and self-contact in nonlinear elasticity. Arch. Rat. Mech. Anal.96, 171–188 (1986)Google Scholar
  14. 14.
    Davie, A.M.: Singular minimizers in the calculus of variations in one dimension. Arch. Rat. Mech. Anal.101, 161–177 (1988)Google Scholar
  15. 15.
    De Giorgi, E.: Un esempio di estremali discontinue per un problema variationale di tipo ellittico. Boll. Un. Mat. Ital.4, 135–137 (1968)Google Scholar
  16. 16.
    Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. Princeton University Press, Princeton, New Jersey 1983Google Scholar
  17. 17.
    Giusti, E., Miranda, M.: Un esempio di soluzioni discontinue per un problema di minimo relativo ad un integrale regolare del calcolo delle variazioni. Boll. Un. Mat. Ital.1, 219–225 (1968)Google Scholar
  18. 18.
    Krasnosel'ski, M.A.: Topological methods in the theory of nonlinear integral equations, New York: Macmillan 1964Google Scholar
  19. 19.
    Lavrentiev, M.: Sur quelques problèmes du calcul des variations. Ann. Math. Pure Appl.4, 7–28 (1926)Google Scholar
  20. 20.
    Maz'ya, V.G.: Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients. Funktsional. Anal. i Prilozhen2, 53–57 (1968)Google Scholar
  21. 21.
    Morrey, C.B.: Multiple integrals in the calculus of variations. (Ed.). Berlin Heidelberg New York: Springer 1966Google Scholar
  22. 22.
    Sivaloganathan, J.: Uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity. Arch. Rat. Mech. Anal.96, 97–136 (1986)Google Scholar
  23. 23.
    Sivaloganathan, J.: A field theory approach to the stability of radial equilibria in nonlinear elastivity. Math. Proc. Camb. Philos. Soc.99, 589–604 (1986)Google Scholar
  24. 24.
    Stuart, C.A.: Radially symmetric cavitation for hyperelastic materials. Ann. Inst. Henri Poincaré, Anal. Nonlinéaire2, 1–20 (1985)Google Scholar
  25. 25.
    Sverák, V.: Regularity properties of deformations with finite energy. Arch. Rat. Mech. Anal.100, 105–206 (1988)Google Scholar
  26. 26.
    Tang, Qi.: Almost everywhere injectivity in nonlinear elasticity (to appear)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Pablo V. Negrón Marrero
    • 1
    • 2
  1. 1.Department of MathematicsUniversity of Puerto RicoRio PiedrasUSA
  2. 2.Department of MathematicsHeriot-Watt UniversityEdinburghScotland

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