Let $n,m\in\mathbb N$, let $\Omega\subset\mathbb R^n$ be open and bounded, let $1<p<\infty$, and let $F:\Omega\times\mathbb R^m\times\mathbb R^{m\times n}\to[0,\infty]$ be a normal integrand, meaning that $x\mapsto F(x,z,P)$ is $\mathcal B(\Omega)$-measurable for every $(z,P)\in\mathbb R^m\times\mathbb R^{m\times n}$ and $(z,P)\mapsto F(x,z,P)$ is lower semicontinuous for $\mathcal L^n$-a.e. $x\in\Omega$. Assume that, for $\mathcal L^n$-a.e. $x\in\Omega$ and every $z\in\mathbb R^m$, the map $P\mapsto F(x,z,P)$ is convex on $\mathbb R^{m\times n}$.

Let $(u_k)_{k\in\mathbb N}\subset W^{1,p}(\Omega;\mathbb R^m)$ and let $u\in W^{1,p}(\Omega;\mathbb R^m)$. Suppose that $u_k\to u$ in $L^p(\Omega;\mathbb R^m)$ and $\nabla u_k\rightharpoonup \nabla u$ in $L^p(\Omega;\mathbb R^{m\times n})$. If

\begin{align*} \sup_{k\in\mathbb N}\int_\Omega F(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x)<\infty, \end{align*}

then

\begin{align*} \int_\Omega F(x,u(x),\nabla u(x))\,d\mathcal L^n(x)\le\liminf_{k\to\infty}\int_\Omega F(x,u_k(x),\nabla u_k(x))\,d\mathcal L^n(x). \end{align*}