Weak extremum

A minimal or maximal value $ J ( \widetilde{y} ) $, attained by a functional $ J ( y) $ on a curve $ \widetilde{y} ( x) $, $ x _ {1} \leq x \leq x _ {2} $, for which one of the following inequalities holds:

$$ J ( \widetilde{y} ) \leq J ( y) \ \textrm{ or } \ \ J ( \widetilde{y} ) \geq J ( y) $$

for all comparison curves $ y ( x) $ situated in an $ \epsilon $- proximity neighbourhood of $ \widetilde{y} ( x) $ with respect to both $ y $ and its derivative:

$$ | y ( x) - \widetilde{y} ( x) | \leq \epsilon ,\ \ | y ^ \prime ( x) - \widetilde{y} {} ^ \prime ( x) | \leq \epsilon . $$

The curves $ \widetilde{y} ( x) $, $ y ( x) $ must satisfy the prescribed boundary conditions.

Since the maximization of $ J ( y) $ is equivalent to the minimization of $ - J( y) $, one often speaks of a weak minimum instead of a weak extremum. The term "weak" emphasizes the fact that the comparison curves $ y ( x) $ satisfy the $ \epsilon $- proximity condition not only on the ordinate but also on its derivative (in contrast to the case of a strong extremum, where the $ \epsilon $- proximity of $ y ( x) $ and $ \widetilde{y} ( x) $ refer only to the former).

By definition, a weak minimum is a weak relative minimum, since the latter gives a minimum among the members of a subset of the whole class of admissible comparison curves $ y( x) $ for which $ J( y) $ makes sense. However, for the sake of brevity, the term "weak minimum" is used for both.

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