Advertisement

General Variational Formulas

  • Mariano Giaquinta
  • Stefan Hildebrandt
Chapter
  • 2.2k Downloads
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 310)

Abstract

In the first two chapters we have investigated the rate of change of variational integrals ℱ with respect to variations of the dependent variables. In particular, we have derived the necessary conditions
$$ \delta \mathcal{F}(u,\varphi ) = 0\quad for\;all\;\varphi \; \in \;C_c^\infty (\Omega ,{\mathbb{R}^\mathbb{N}})$$
and
$$ {L_F}(u) = 0$$
for extremizers u of ℱ, and suitable modifications of these conditions were shown to hold for constrained extremizers.

Keywords

Vector Field Euler Equation Infinitesimal Generator Fundamental Lemma Weak Minimizer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    In honour of Emmy Noether; cf. 3, 3 and 3, 4.Google Scholar
  2. 2.
    Note that, by definition, weak extremals are of class C1. Later we shall admit weak extremals which are much less regular. Then the regularity theorem does not always hold.Google Scholar
  3. 3.
    It is throughout used in his celebrated lecture notes [1] Vol. 7; cf., in particular, pp. 107–108.Google Scholar
  4. 4.
    See the Supplement for the definition of K g . Compare also Bolza [3], p. 210.Google Scholar
  5. 5.
    Often, one adds the factor 1/2, cf. 2,4 3 math. We have dropped it to simplify the following formulas.Google Scholar
  6. 6.
    Cf. e.g. Courant-Hilbert [1–4].Google Scholar
  7. 7.
    To spare the reader any confusion, we have to remark that in the literature one often finds the notations δu =φ or δ*u = φ for the “variation of u with respect to fixed arguments”, and δu = δu + Du • δx or δu = δ*u + Du • δx for the “total” variation of u. Similarly, one finds ∂ℱ =Φ’(0) instead of the notation (8). Our notation has the advantage that it integrates the previous definition of ∂ℱ:Google Scholar
  8. 8.
    see 2 math of this section.Google Scholar
  9. 9.
    see 2 math of this section.Google Scholar
  10. 10.
    In his Königsberg lectures [4], pp. 198–211, elliptic coordinates in Rn are dealt with in full detail. For historical references, cf. Jacobi [3], Vol. 2, pp. 59–63.Google Scholar
  11. 11.
    For a presentation within the framework of differential geometry we refer the reader to the Supplement.Google Scholar
  12. 12.
    The convention about the sign of Δ is not uniform, many authors prefer the opposite sign.Google Scholar
  13. 13.
    An essay on the cohesion of fluids, Phil. Trans. Roy. Soc. London 95, 65–87 (1805).Google Scholar
  14. 14.
    Mémoire sur la courbure des surfaces, Mém. de Math. Phys. (sav. etrang.) de l’Acad. 10 (1785), 447–550 (lu 1776).Google Scholar
  15. 15.
    (E385) Institutionum calculi integralis, Vol. 3, Petersburg 1770, appendix; cf. also: A. Kneser, Variationsrechnung, Enzykl. math. Wiss., Vol. 2.1, IIA8, pp. 575–576.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

Personalised recommendations