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The First Variation

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Calculus of Variations I

Part of the book series: Grundlehren der mathematischen Wissenschaften ((GL,volume 310))

Abstract

In this chapter we shall develop the formalism of the calculus of variations in simple situations. After a brief review of the necessary and sufficient conditions for extrema of ordinary functions on ℝn, we investigate in Section 2 some of the basic necessary conditions that are to be satisfied by minimizers of variational integrals

$$ \mathcal{F}(u) = \int_\Omega {F(x,u(x),Du(x)dx)} $$
((1))

.

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References

  1. The reader may wonder why we operate with (math) instead of (math). Here we follow the custom of the theory of distributions. It has the advantage that one need not distinguish between “test functions” for differential equations of different orders. Note that, under our assumptions, (7) implies that δℱ(u,φ) holds for all (math).

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Authors and Affiliations

  1. Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56100, Pisa, Italy

    Mariano Giaquinta

  2. Mathematisches Institut, Universität Bonn, Wegelerstr. 10, 53115, Bonn, Germany

    Stefan Hildebrandt

Authors
  1. Mariano Giaquinta
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  2. Stefan Hildebrandt
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© 2004 Springer-Verlag Berlin Heidelberg

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Giaquinta, M., Hildebrandt, S. (2004). The First Variation. In: Calculus of Variations I. Grundlehren der mathematischen Wissenschaften, vol 310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03278-7_1

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  • DOI: https://doi.org/10.1007/978-3-662-03278-7_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08074-6

  • Online ISBN: 978-3-662-03278-7

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