Past exam of the calculus of variations course of the University of Oxford / 2011 / 1 / b / ii

The integrand is nonnegative, so $\inf I\ge 0$. For each positive integer $n$, setThis is a continuous piecewise affine sawtooth with zero endpoints, $0\le u_n\le 1/n$, and $u_n^{\prime }=\pm 2$ almost everywhere. It belongs to $W^{1,\infty }$, and $(u_n^{\prime }{)}^6=64$ almost everywhere. ThereforeTogether with nonnegativity this proves $\boxed{\underset{\mathcal{A}}{\inf }I=0}$. The functions converge uniformly to zero while their derivatives retain magnitude two; small amplitude does not force small gradient.