= Solution
Suppose $u_k\rightharpoonup u$ in $H^1(\Omega)$. Then $Du_k\rightharpoonup Du$ in $L^2(\Omega;\mathbb R^n)$. Since $F$ is a differentiable <convex function>, its supporting inequality is
$$
F(p)\geq F(q)+DF(q)\cdot(p-q).
$$
Take $p=Du_k(x)$, $q=Du(x)$ and integrate. Part (i) shows that every energy is finite and that the fixed field $DF(Du)$ belongs to $L^2$, so
$$
\mathcal F(u_k)\geq\mathcal F(u)+\int_\Omega DF(Du)\cdot(Du_k-Du).
$$
The last integral tends to zero by the definition of <weak convergence> in $L^2$. Therefore
$$
\boxed{\mathcal F(u)\leq\liminf_{k\to\infty}\mathcal F(u_k).}
$$
This 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.
Back to article page
