Another Counterexample to Lower Semicontinuity in Calculus of Variations

An example is shown of a functional $$F(u)=\int_{I}f(u,u')\,dt$$ which is not lower semicontinuous with respect to $L^1$-convergence. The function $f$ is nonnegative, continuous and strictly convex in the second variable for each $u \in {\mathbb R}^n$.