Second Variation, Excess Function, Convexity

  • Mariano Giaquinta
  • Stefan Hildebrandt
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 310)


If (u) is a real valued function of a real variable which is of class C 2 on ℝ, then, besides ′(u0) = 0, also the condition ″(u0) ≥ 0 is necessary for u 0 being a local minimizer of . Moreover, the conditions
$$ \mathcal{F}(u) = 0\int_\Omega F (x,u,Du)\;dx$$
are sufficient for u 0 furnishing a local minimum.


Minimum Property Unique Minimizer Admissible Function Jacobi Operator Excess Function 
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  1. 1.
    The first to point out this fact were Weierstrass in his lectures on the calculus of variations, and Scheeffer in his very influentual paper [3] from 1886. Cf. also Bolza’s treatise [3], Chapter 3, and the historical account in H.H. Goldstine [1], Chapter 5 (in particular 5.5 and 5.11).Google Scholar
  2. 2.
    One might justly object that the C1-topology is stronger than the C0-topology. Thus it seems more appropriate to denote weak and strong neighbourhoods as narrow and wide neighbourhoods respectively. Google Scholar
  3. 3.
    Cf. Carathéodory [16], Vol. 5, pp. 165–174, and Euler [1], Ser. I, Vol. 24, pp. VIII-LXIII for a list of variational problems studied in Euler [2].Google Scholar
  4. 4.
    Methodus inveniendi [2], Chapter IV, Proposition I. Cf. also Carathéodory [16], Vol. 5, p. 125, and Goldstine [1], pp. 84–92.Google Scholar
  5. 5.
    Cf. Carathéodory [16], Vol. 5, p. 118, and Euler, Opera omnia l, Vol. 24.Google Scholar
  6. 6.
    Actually, this time-honoured trick has already been used by Legendre [1] in 1786 to prove the positivity of the second variation. This directly leads to Jacobi’s theory for n = N = 1; cf. Chapter 5.Google Scholar
  7. 7.
    Actually, intuition for functions of more than one variable — and especially for variational problems — can be elusive. For instance, there are functions f ∈ C∞(lR2) with a unique critical point which, in addition, is a local minimizer, but which nevertheless have no global minimizer. An example of this kind is provided by (math)Google Scholar
  8. 8.
    Mémoire sur la manière de distinguer les maxima des minima dans le calcul des variations, Mém. de l’acad. sci. Paris (1786) 1788, 7–37.Google Scholar
  9. 9.
    Zur Theorie der Variations-Rechnung und der Differential-Gleichungen, Crelle’s Journal f. d. reine u. angew. Math. 17, 68–82 (1837); cf. Werke [3], Vol. 4, pp. 39–55.Google Scholar
  10. 10.
    Vorlesungen über Variationsrechnung, bearbeitet von Rudolf Rothe, Math. Werke [1], Vol. 7, Leipzig, 1927. 11 Ges. Math. Schriften [16], Vol. 5, pp. 343–347.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Mariano Giaquinta
    • 1
  • Stefan Hildebrandt
    • 2
  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany

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