\begin{align*}
\mathcal A=\{u\in W^{1,1}(0,1):\text{ the one-dimensional Sobolev representative of }u\text{ satisfies }u(0)=0\text{ and }u(1)=1\}.
\end{align*}
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Define $I:\mathcal A\to[0,\infty]$ as follows. If the [measurable function](/page/Measurable%20Function) $x\mapsto (u(x)^3-x)^2|u'(x)|^6$ belongs to $L^1(0,1)$, set
Assume the classical one-dimensional Manià lower-bound estimate for this endpoint class, namely $c_M>0$. Then there exists $u_0\in\mathcal A$ such that $I[u_0]=0$, and hence
The integrand $L:[0,1]\times\mathbb R\times\mathbb R\to[0,\infty)$ given by $L(x,z,\xi)=(z^3-x)^2|\xi|^6$ is continuous, nonnegative, and convex in $\xi$.