Solution
= Solution
A <weak local minimizer of a variational integral> is an admissible $\bar u$ for which there is $\rho>0$ such that
$$
\boxed{I(v)\geq I(\bar u)\quad\text{whenever }v\in\mathcal A,\ \|v-\bar u\|_{W^{1,\infty}}<\rho.}
$$
The <Sobolev norm> controls both the function and its first <weak derivative>. The boundary values remain fixed because both functions are admissible.