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The Direct Method of the Calculus of Variations

  • Jean Mawhin
  • Michel Willem
Part of the Applied Mathematical Sciences book series (AMS, volume 74)

Abstract

A real function φ of a real variable which is bounded below on the real line needs not to have a minimum, as it is clear from the example of the exponential function. If we call minimizing sequence for φ any sequence (a k ) such that
$$ \varphi ({a_k}) \to \inf \varphi $$
as k → ∞, a necessary condition for the real number a to be such that
$$ \varphi (a) \to \inf \varphi $$
is that φ has a minimizing sequence which converges to a (take a k = a for all integers k). Without suitable continuity assumptions on φ this condition will not be sufficient, as shown by the example of the function φ defined by φ(x) = |x| for x ≠ 0 and φ(0) = 1, which does not achieve its infimum 0 although all its minimizing sequences converge to zero. In order that the limit a of a convergent minimizing sequence be such that φ(a) = inf φ, we have to impose that
$$ \mathop {\lim }\limits_{k \to \infty } \varphi ({a_k}) \ge \varphi (a) $$

Keywords

Convex Function Reflexive Banach Space Fundamental Lemma Weak Derivative Bibliographical Note 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Jean Mawhin
    • 1
  • Michel Willem
    • 1
  1. 1.Institut de Mathematique Pure et AppliqueeLouvain-la-NeuveBelgium

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