= Oscillating minimizing sequence with no minimizer
{title2=$u_N(x)=2\operatorname{dist}(x,N^{-1}\mathbb Z)$}
A <minimizing sequence> can converge uniformly while its <derivatives> keep oscillating between distinct energy wells. For the <functional> $I(u)=\int_0^1[u^4+(u'^4-16)^4]$, with zero endpoint values and <Lipschitz> admissible functions, the triangular waves displayed above have slopes $\pm2$ <almost everywhere> and height $1/N$. Direct integration gives $I(u_N)=1/(5N^4)$. Nonnegativity therefore makes the <infimum> zero. Attainment would require both $u=0$ and $|u'|=2$ <almost everywhere>, an impossibility. The uniform limit is zero but its energy is $16^4$; thus the energy fails to be <lower semicontinuous> for this convergence. The example explains why control of <function> values alone cannot replace suitable <convexity> in the <derivative> variable in the <direct method in the calculus of variations>.
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