Weak extremum

A minimal or maximal value $J(\tilde{y})$, attained by a functional $J(y)$ on a curve $\tilde{y}(x)$, $x_1\le x\le x_2$, for which one of the following inequalities holds:

$$J(\tilde{y})\le J(y)\text{ or }J(\tilde{y})\ge J(y)$$

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

$$|y(x)-\tilde{y}(x)|\le ϵ,|y^{\prime }(x)-\tilde{y}{}^{\prime }(x)|\le ϵ.$$

The curves $\tilde{y}(x)$, $y(x)$ 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 $ϵ$- proximity condition not only on the ordinate but also on its derivative (in contrast to the case of a strong extremum, where the $ϵ$- proximity of $y(x)$ and $\tilde{y}(x)$ 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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