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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.

For a scalar fixed-endpoint problem on a bounded interval, suppose the continuous integrand $f(x,y,p)$ is convex in $p$ and satisfies $f(x,y,p)\ge \Theta (|p|)-C$, where $\Theta$ is nonnegative convex and $\Theta (r)/r\to \infty$. If there is one admissible absolutely continuous function of finite energy, a global minimizer exists in $W^{1,1}$.

The lower bound gives uniform integrability of derivatives in a minimizing sequence. Fixed endpoint values then give boundedness and equicontinuity of the functions. Extract uniform convergence of the functions and weak $L^1$ convergence of their derivatives; the limit retains the endpoint values. Convexity in the derivative gives weak lower semicontinuity of convex gradient energies, so the limit attains the infimum. Strict convexity of the full integrand on a convex admissible class additionally gives uniqueness.