 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 $I(v)\ge I(u)$ whenever the Sobolev norm $\Vert v-u{\Vert }_{W^{1,\infty }}$ 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.