# 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 .

## Preview

### 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
3. 3.
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.Google Scholar
4. 4.
This is a consequence of the unique solvability of the Cauchy problem for a regular system of ordinary differential equations.Google Scholar
5. 5.
For general results about the theory of ordinary differential equations see e.g. Hartman [1] or Coddington-Levinson [1].Google Scholar
6. 6.
Cf. Carathéodory [10], p. 295, and Bolza [3], §47, pp. 357–364.Google Scholar
7. 7.
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}.Google Scholar
8. 8.
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.Google Scholar
9. 9.
Sturm, ℱMémoire sur les équations différentielles du second ordre, Journal de Liouville 1, 106–186 (1836).Google Scholar
10. 10.
Bull. American Math. Soc. ℱ4, 295–313 & 365–376 (1898); cf. also Kamke [3], pp. 125–128, and 260–261.Google Scholar