Abstract
If ℱ(u) is a real-valued function of real variables u ∈ℝn which is of class C 2, then the positive definiteness of its Hessian matrix at a stationary point u is sufficient to guarantee that u is a relative minimizer. In other words, the assumption D 2 ℱ(u) > 0 for the stationary point u implies that
for all v in a sufficiently small neighbourhood of u in ℝn.
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References
One can even prove that ℱ03BB;1(Ω’)>0 holds if the measure of ℱ03A9;’ is sufficiently small.
Cf. Morrey [1], Chapter 6.
Cf. Leis [1], pp. 64–69, 217–219; Hörmander [1], pp. 224–229, and [2], Vol. III, Section 17.2, Vol. IV, Sections 28.1–28.4.
This is a consequence of the unique solvability of the Cauchy problem for a regular system of ordinary differential equations.
For general results about the theory of ordinary differential equations see e.g. Hartman [1] or Coddington-Levinson [1].
Cf. Carathéodory [10], p. 295, and Bolza [3], §47, pp. 357–364.
We recall that the envelope ℱ03B4; of the family φ(x, α) is defined by ℱ03B4; := {(x, z): there is an a such that ℱz = φℱ(x, αℱ) and φα ℱ(x,α) = 0}.
Since our present notation is inadequate, the reader may get the impression that the following discussion only applies to suitably small pieces of ℱS This, however, is not the case.
Sturm, ℱMémoire sur les équations différentielles du second ordre, Journal de Liouville 1, 106–186 (1836).
Bull. American Math. Soc. ℱ4, 295–313 & 365–376 (1898); cf. also Kamke [3], pp. 125–128, and 260–261.
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© 2004 Springer-Verlag Berlin Heidelberg
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Giaquinta, M., Hildebrandt, S. (2004). Weak Minimizers and Jacobi Theory. In: Calculus of Variations I. Grundlehren der mathematischen Wissenschaften, vol 310. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03278-7_5
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DOI: https://doi.org/10.1007/978-3-662-03278-7_5
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