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
and
for extremizers u of ℱ, and suitable modifications of these conditions were shown to hold for constrained extremizers.
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References
In honour of Emmy Noether; cf. 3, 3 and 3, 4.
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.
It is throughout used in his celebrated lecture notes [1] Vol. 7; cf., in particular, pp. 107–108.
See the Supplement for the definition of K g . Compare also Bolza [3], p. 210.
Often, one adds the factor 1/2, cf. 2,4 3 math. We have dropped it to simplify the following formulas.
Cf. e.g. Courant-Hilbert [1–4].
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:
see 2 math of this section.
see 2 math of this section.
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 For historical references, cf.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.
For a presentation within the framework of differential geometry we refer the reader to the Supplement.
The convention about the sign of Δ is not uniform, many authors prefer the opposite sign.
An essay on the cohesion of fluids, Phil. Trans. Roy. Soc. London 95, 65–87 (1805).
Mémoire sur la courbure des surfaces, Mém. de Math. Phys. (sav. etrang.) de l’Acad. 10 (1785), 447–550 (lu 1776).
(E385) Institutionum calculi integralis, Vol. 3, Petersburg 1770, appendix; cf. also: A. Kneser, Variationsrechnung, Enzykl. math. Wiss., Vol. 2.1, IIA8, pp. 575–576.
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Giaquinta, M., Hildebrandt, S. (2004). General Variational Formulas. In: Calculus of Variations I. Grundlehren der mathematischen Wissenschaften, vol 310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03278-7_3
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DOI: https://doi.org/10.1007/978-3-662-03278-7_3
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