Past exam of the partial differential equations for pure and applied mathematicians and calculus of variations course of the University of Oxford / 2007 / B / 8 / a / iii / Solution

Write $l={\inf }_{\mathcal{A}}I$. Since the limit is admissible, the assumed lower-semicontinuity inequality gives $l\le I(u)\le \mathrm{liminf}I(u_{j_k})=l$. Thus $I(u)=l$ and $u$ is a global minimizer.

The extra integrand hypothesis in the Tonelli existence theorem for one dimension is convexity in the derivative variable for each fixed $(x,u)$. Together with the stated continuity and coercive lower bound it gives the lower-semicontinuity property along the uniform-function/weak-derivative convergence just proved.