= Solution
For any admissible $v$, put $q=v-w$. At endpoints $q=0$. On the <contact set of an obstacle problem>[contact set], $w=f$ and $v\geq f$ imply $q\geq0$, while away from contact the reaction $r=p-Tw''$ is zero. Hence $rq\geq0$ everywhere. <Integration by parts> gives
$$
\begin{aligned}
T\int_{-1}^1w'(v'-w')\,dx+\int_{-1}^1p(v-w)\,dx
&=\int_{-1}^1(p-Tw'')(v-w)\,dx\\
&=\int_{-1}^1r(v-w)\,dx\geq0.
\end{aligned}
$$
Rearranging proves the requested <variational inequality>. Piecewise integration is legitimate because the <slope> is continuous at the contact boundary, so there are no omitted point-force terms.
Back to article page
