 Strong local minimizer of a variational integral

For a fixed-endpoint variational integral, an admissible function $u$ is a strong local minimizer if $I(v)\ge I(u)$ for all admissible $v$ whose uniform norm distance from $u$ is sufficiently small. Derivatives can be far apart. Hence strong local minimality implies weak local minimality, while a global minimizer implies both. Small narrow bumps and rapid low-amplitude oscillations explain why the reverse implication can fail.