Analytical Background

  • Sören BartelsEmail author
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 47)


The calculus of variations provides an attractive mathematical framework for describing many phenomena in continuum mechanics. In this chapter model problems, including bending thin elastic objects as well as elastoplastic material behavior, are formulated within this framework, and general concepts such as the direct method in the calculus of variations and gradient flows that imply their well-posedness are discussed. The definitions and most important properties of Sobolev and Sobolev–Bochner spaces are provided.


  1. 1.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces. MPS/SIAM Series on Optimization, vol. 6. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2006)Google Scholar
  2. 2.
    Brézis, H.R.: Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Publishing Co., Amsterdam (1973). North-Holland Mathematics Studies, No. 5. Notas de Matemática (50)Google Scholar
  3. 3.
    Ciarlet, P.G.: Mathematical Elasticity. Vol. I: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, vol. 20. North-Holland Publishing Co., Amsterdam (1988)Google Scholar
  4. 4.
    Dacorogna, B.: Direct Methods in the Calculus of Variations. Applied Mathematical Sciences, vol. 78, 2nd edn. Springer, New York (2008)Google Scholar
  5. 5.
    Eck, C., Garcke, H., Knabner, P.: Mathematische Modellierung, 2nd edn. Springer Lehrbuch. Springer, Berlin (2011)Google Scholar
  6. 6.
    Evans, L.C.: Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010)Google Scholar
  7. 7.
    Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics, vol. 153, 2nd edn. Birkhäuser/Springer Basel AG, Basel (2013)Google Scholar
  8. 8.
    Ružička, M.: Nichtlineare Funktional Analysis. Springer, Berlin-Heidelberg-New York (2004)Google Scholar
  9. 9.
    Salsa, S., Vegni, F.M.G., Zaretti, A., Zunino, P.: A Primer on PDEs. Unitext, vol. 65, italian edn. Springer, Milan (2013)Google Scholar
  10. 10.
    Struwe, M.: Variational Methods, 4th edn. Springer, Berlin (2008)Google Scholar
  11. 11.
    Temam, R., Miranville, A.: Mathematical Modeling in Continuum Mechanics, 2nd edn. Cambridge University Press, Cambridge (2005)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

Personalised recommendations