Weak local minimizer of a variational integral
= Weak local minimizer of a variational integral
For a fixed-endpoint integral on $W^{1,\infty}(a,b)$, an admissible function $u$ is a weak local minimizer if $I(v)\geq I(u)$ whenever the <Sobolev norm> $\|v-u\|_{W^{1,\infty}}$ is sufficiently small. Both function and derivative must be close. “Weak” here labels the variational neighborhood convention, not the <weak topology> of a <Banach space>.