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.

With fixed endpoint values on a finite interval, the displayed derivative coercivity makes a finite-energy strong local minimizer of a variational integral also a local minimizer in $L^1$. Otherwise choose better competitors converging in $L^1$. The upper energy bound controls their derivatives in $L^p$, and a common endpoint controls their function norms. Compact Sobolev embedding gives a uniformly convergent subsequence; its limit is the $L^1$ limit. This contradicts strong local minimality. Convexity of the integrand is unnecessary for this neighborhood-transfer argument.