Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2007 / B / 5 / a / iii / Solution

For a zero-trace $\varphi$, the curve $u+t\varphi$ remains admissible. When $|t|$ is small it is within the weak-minimum neighborhood in $W^{1,\infty }$, so its scalar energy has a minimum at zero. The preceding derivative therefore vanishes. Thus $DI(u)\varphi =0$ for every zero-trace variation.