 Solution

Suppose $u_k\rightharpoonup u$ in $H^1(\Omega )$. Then $D{u}_k\rightharpoonup Du$ in $L^2(\Omega ;{\mathbb{R}}^n)$. Since $F$ is a differentiable convex function, its supporting inequality isTake $p=D{u}_k(x)$, $q=Du(x)$ and integrate. Part (i) shows that every energy is finite and that the fixed field $DF(Du)$ belongs to $L^2$, soThe last integral tends to zero by the definition of weak convergence in $L^2$. ThereforeThis is weak lower semicontinuity of convex gradient energies. It uses a fixed supporting gradient at the limit; weak convergence alone would not justify passing pointwise through the nonlinear integrand.