= Solution
The small-<slope> excess length is $\int(w')^2/2$, so the first term is the work needed against uniform <tension>. The second is the <potential energy> of the downward distributed load. Their sum is total <potential energy> subject to the obstacle <constraint>.
Expand the difference exactly:
$$
U[v]-U[w]=T\int w'(v'-w')\,dx+\int p(v-w)\,dx+\frac T2\int(v'-w')^2\,dx.
$$
The first two integrals together are nonnegative by part (b), and the square is nonnegative. Thus
$$
\boxed{U[w]\leq U[v]\quad\text{for every admissible }v.}
$$
For $T>0$ this also proves <uniqueness>: equality for two <minimizers> forces identical derivatives, and the fixed endpoints remove their additive constant. The <convex> admissible set and strictly <convex> energy make the <variational inequality> a global minimization condition.
Back to article page
