 Weak lower semicontinuity of convex gradient energies

Suppose the domain has finite measure and $F\in C^1$ is convex, with $|F(p)|\le C(1+|p{|}^2)$ and $|DF(p)|\le C(1+|p|)$. The supporting inequality $F(p)\ge F(q)+DF(q)\cdot (p-q)$, integrated with $q=Du$, gives $\int F(D{u}_k)\ge \int F(Du)+\int DF(Du)\cdot (D{u}_k-Du)$. The last term tends to zero by weak convergence of the gradients and the fixed $L^2$ test field $DF(Du)$. This proves weak lower semicontinuity without requiring pointwise convergence of gradients.