Past exam of the calculus of variations course of the University of Oxford / 2009 / 1 / b

An admissible $\bar{u}$ is a global minimizer ifIn particular it attains the infimum; no closeness condition on competing admissible functions is imposed.

The integrand is nonnegative. Define the Lipschitz triangular wavesThey have zero endpoint values, slope $\pm 2$ almost everywhere, and height $1/N$. Hence $(u_N^{\prime 4}-16{)}^4=0$ almost everywhere. On each of the $N$ equal cells, direct integration of the fourth power of the two affine pieces gives $1/(5N^5)$. ThereforeThis is an oscillating minimizing sequence with no minimizer.

If an admissible $u$ attained the infimum, nonnegativity and $I(u)=0$ would force $u^4=0$ almost everywhere, so its continuous Sobolev representative would be identically zero. Its derivative would then vanish almost everywhere, but the other integrand term would equal $16^4>0$. This is impossible. Equivalently, zero energy would simultaneously require $u=0$ and $|u^{\prime }|=2$ almost everywhere. Thus