Weak Minimizers and Jacobi Theory

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

Abstract

If (u) is a real-valued function of real variables u ∈ℝ n 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 (u) > 0 for the stationary point u implies that
$$\mathcal{F}(u) \leqslant \mathcal{F}(v) $$
for all v in a sufficiently small neighbourhood of u in ℝ n .

Keywords

Manifold Radon Dition Flint 

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References

  1. 1.
    One can even prove that ℱ03BB;1(Ω’)>0 holds if the measure of ℱ03A9;’ is sufficiently small.Google Scholar
  2. 2.
    Cf. Morrey [1], Chapter 6.Google Scholar
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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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