= Weak lower semicontinuity of convex gradient energies
{title2=$u_k\rightharpoonup u\text{ in }H^1\Longrightarrow \int F(Du)\leq\liminf_k\int F(Du_k)$}
Suppose the domain has finite measure and $F\in C^1$ is convex, with $|F(p)|\leq C(1+|p|^2)$ and $|DF(p)|\leq C(1+|p|)$. The supporting inequality $F(p)\geq F(q)+DF(q)\cdot(p-q)$, integrated with $q=Du$, gives $\int F(Du_k)\geq\int F(Du)+\int DF(Du)\cdot(Du_k-Du)$. 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.
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