Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Direct Method of Calculus of Variations in Elasticity

  • Alessandro Della Corte
  • Francesco dell’Isola
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_174-1


Theorems on the existence of minimizers for functionals defined on Banach spaces, and related approximation methods, applied to minimization problems arising in elasticity theory.


Variational methods are a powerful tool in elasticity and in fact the only known approach able to guarantee sufficient generality in the treatment of problems arising in hyperelasticity (see the corresponding entry), in the asymptotic derivation of two-dimensional elastic models (Ciarlet, 1988, 1997), and in many other cases in continuum mechanics (Pedregal, 2000; Fonseca, 1987; dell’Isola and Placidi, 2011). Variational problems take usually the form of a minimization problem that can be described as follows: we search the minimum of a functional F(u) defined on a subset S of a Banach space B (i.e., a normed, complete vector space) and taking values in [−, +]. Direct method provides sufficient conditions on S, B, and F for the existence of a minimizer \(\tilde {u}\in S\)

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  1. Alaoglu L (1940) Weak topologies of normed linear spaces. Ann Math 41:252–267MathSciNetCrossRefMATHGoogle Scholar
  2. Axelsson O, Barker VA (2001) Finite element solution of boundary value problems: theory and computation. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  3. Ball JM (1976) Convexity conditions and existence theorems in nonlinear elasticity. Arch Ration Mech Anal 63(4):337–403MathSciNetCrossRefMATHGoogle Scholar
  4. Ball JM, Mizel VJ (1987) One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. In: Analysis and thermomechanics. Springer, Berlin/Heidelberg, pp 285–348CrossRefGoogle Scholar
  5. Banach S (1932) Théorie des opérations linéairesGoogle Scholar
  6. Belloni M (1995) Interpretation of lavrentiev phenomenon by relaxation: the higher order case. Trans Am Math Soc 347:2011–2023MathSciNetCrossRefMATHGoogle Scholar
  7. Berkovitz LD (1974) Lower semicontinuity of integral functionals. Trans Am Math Soc 192:51–57MathSciNetCrossRefMATHGoogle Scholar
  8. Bouchitté G, Braides A, Buttazzo G (1995) Relaxation results for some free discontinuity problems. Journal fur die Reine und Angewandte Mathematik 458:1–18MathSciNetMATHGoogle Scholar
  9. Ciarlet PG (1988) Mathematical elasticity. vol. I, volume 20 of studies in mathematics and its applications. North-Holland, AmsterdamMATHGoogle Scholar
  10. Ciarlet PG (1997) Mathematical elasticity, vol. II: theory of plates, volume 27 of studies in mathematics and its applications. North-Holland, AmsterdamMATHGoogle Scholar
  11. Dacorogna B (2007) Direct methods in the calculus of variations, vol 78. Springer, New YorkMATHGoogle Scholar
  12. Della Corte A, dell’Isola F, Esposito R, Pulvirenti M (2017) Equilibria of a clamped euler beam (elastica) with distributed load: large deformations. Math Models Methods Appl Sci 27(8):1391–1421Google Scholar
  13. dell’Isola F, Placidi L (2011) Variational principles are a powerful tool also for formulating field theories. In: Variational models and methods in solid and fluid mechanics. Springer, Vienna, pp 1–15Google Scholar
  14. Fonseca I (1987) Variational methods for elastic crystals. Arch Ration Mech Anal 97(3):189–220MathSciNetCrossRefMATHGoogle Scholar
  15. Fonseca I, Leoni G (2007) Modern methods in the calculus of variations: Lˆ p spaces. Springer, New YorkMATHGoogle Scholar
  16. Galerkin BG (1915) Series solution of some problems of elastic equilibrium of rods and plates. Vestn Inzh Tekh 19:897–908Google Scholar
  17. Gelfand IM, Fomin SV (1963) Richard A. Silverman (ed) Calculus of variations. Prentice-Hall, Englewood CliffsGoogle Scholar
  18. Kantorovich L, Krylov V (1962) Approximate methods of advanced analysis. Gos Izd Fiz-Math Lit, Leningrad 560Google Scholar
  19. McShane E et al (1940) Necessary conditions in generalized-curve problems of the calculus of variations. Duke Math J 7(1):1–27MathSciNetCrossRefMATHGoogle Scholar
  20. Mikhlin SG (1957) Variatsionnye metody v matematicheskoi fizike. GostekhizdatGoogle Scholar
  21. Morrey CB (1952) Quasi-convexity and the lower semicontinuity of multiple integrals. Pac J Math 2(1):25–53MathSciNetCrossRefMATHGoogle Scholar
  22. Pedregal P (2000) Variational methods in nonlinear elasticity. SIAM, PhiladelphiaCrossRefMATHGoogle Scholar
  23. Ritz W (1908) Uber eine neue methode zur losung gewisser variations-probleme der mathematischen physik. Journal fuur die reine und angewandte Mathematik 135(1):1–61MATHGoogle Scholar
  24. Roubicek T (1997) Relaxation in optimization theory and variational calculus, vol 4. Walter de Gruyter, BerlinCrossRefMATHGoogle Scholar
  25. Rudin W (1991) Functional analysis. International series in pure and applied mathematics. McGraw-Hill, New YorkGoogle Scholar
  26. Smirnov VI (2014) A course of higher mathematics, vol 62. Elsevier, BurlingtonGoogle Scholar
  27. Sobolev SL (1950) Some applications of functional analysis in mathematical physics, vol 90. American Mathematical Society, ProvidenceGoogle Scholar
  28. Tonelli L (1911) Sui massimi e minimi assoluti del calcolo delle variazioni. Rendiconti del Circolo Matematico di Palermo (1884–1940) 32(1):297–337CrossRefMATHGoogle Scholar
  29. Tonelli L (1920) La semicontinuita nel calcolo delle variazioni. Rendiconti del Circolo Matematico di Palermo (1884–1940) 44(1):167–249CrossRefMATHGoogle Scholar
  30. Young LC (1937) Generalized curves and the existence of an attained absolute minimum in the calculus of variations. Comptes Rendus de la Société des Sci et des Lettres de Varsovie 30:212–234MATHGoogle Scholar
  31. Young LC (1938) Necessary conditions in the calculus of variations. Acta Mathematica 69(1):229–258MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  • Alessandro Della Corte
    • 1
    • 2
  • Francesco dell’Isola
    • 2
    • 3
  1. 1.DIMAUniversity of Rome La SapienzaRomeItaly
  2. 2.International Research Center M&MoCSUniversity of L’AquilaL’AquilaItaly
  3. 3.DISGUniversity of Rome La SapienzaRomeItaly

Section editors and affiliations

  • Francesco dell’Isola
    • 1
    • 2
  1. 1.DISGUniversity of Rome La SapienzaRomeItaly
  2. 2.International Research Center M&MoCSL’AquilaItaly