Extrema of Differentiable Functions

Part of the Texts and Monographs in Physics book series (TMP)


In this chapter we would like to investigate how one can calculate, for a variational problem, a minimising point whose existence and possibly also whose uniqueness (in the sense of Chap. 1) have been established. It is not possible, with the theorems we proved in Chap. 1 for the existence of an extremum of a functional f: M → ℝ, (where M is an open subset of a Banach space E), to find those points at which this functional attains, for example, its minimum. However, it is possible to do this with the aid of differential calculus in Banach spaces, which we introduced in Chap. 2, taking the approach used in the well-known case of differentiable functions on the real axis.


Banach Space Open Subset Extremal Point Differentiable Function Real Hilbert Space 
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  1. 3.1
    Vainberg, M.M.: Variational methods for the study of nonlinear operators. Holden Day, London 1964zbMATHGoogle Scholar
  2. 3.2
    Dieudonné, J.: Foundations of modern analysis. Academic Press, New York 1969zbMATHGoogle Scholar

Further Reading

  1. Courant, R., Hilbert, D.: Methods of mathematical physics. Wiley-Interscience, New York 1966Google Scholar
  2. Velte, W.: Direkte Methoden der Variationsrechnung. Teubner, Stuttgart 1976zbMATHGoogle Scholar
  3. Young, L.C.: Calculus of variation and optimal control theory. Saunders, Philadelphia 1969Google Scholar
  4. Gelfand, I.M., Fomin, S.V.: Calculus of variations. Prentice-Hall, Englewood Cliffs 1963Google Scholar
  5. Funk, P.: Variationsrechnung und ihre Anwendung in Physik und Technik. Springer, Berlin Heidelberg 1962zbMATHGoogle Scholar
  6. Morse, M.: The calculus of variations in the large. Am. Math. Soc., Providence 1934Google Scholar
  7. Fucik, S., Nenas, J., Soucek, V.: Einführung in die Variationsrechnung. Teubner, Leipzig 1977zbMATHGoogle Scholar
  8. Jaffe, A., Taubes, C.: Vortices and monopoles. Birkhäuser, Boston 1980zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  1. 1.Theoretische Physik, Fakultät für PhysikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsUniversity of Cape TownRondeboschSouth Africa

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