Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2008 / B / 6 / a / i / Solution

A weak local minimizer of a variational integral is an admissible $\bar{u}$ for which some $\rho >0$ gives $I(v)\ge I(\bar{u})$ for all admissible $v$ with $\Vert v-\bar{u}{\Vert }_{W^{1,\infty }}<\rho$.