ما معنى Variational inequality, free-boundary problem, smooth pasting؟

The display above is a variational inequality. Written as “solve $\mathcal LV = rV$ on an unknown region $D$ with $V = g$ on its boundary”, it is a free-boundary problem: the boundary $\partial D$ is part of the solution. The extra condition that fixes it, $V' = g'$ on $\partial D$ where $g$ is smooth, is smooth pasting (or smooth fit).