Let $\Omega$ be a bounded domain (smooth if necessary) and let $J:H^1(\Omega) \times H^1\_0(\Omega) \to \mathbb{R}$ be defined by


$$J(u,v) = \int\_\Omega f(u)|\nabla v|^2$$
where $f\colon \mathbb{R} \to \mathbb{R}$ is a smooth function, bounded above and below away from zero, which can make as nice as necessary.


Under what conditions on $f$ do I get that $J$ is weakly lower semicontinuous? 


Obviously if $f \equiv 1$ then it is true, but what about the more general case?



Weak lower semicontinuity of functional with two arguments
This functional is sequentially weakly lower semicontinuous under fairly mild assumptions on $f$.
We need that $f$ is non-negative, continuous and bounded from above.


Let $u\_n \rightharpoonup u$ in $H^1(\Omega)$ and $v\_n \rightharpoonup v$ in $H\_0^1(\Omega)$.
Rellich-Kondrachov implies that $u\_n \to u$ in $L^2(\Omega)$ (we might need a mild assumption on $\Omega$ here).
Hence, $f(u\_n) \to f(u)$ (along a subsequence) a.e.


For $\varepsilon > 0$ we can use Egorov's theorem to get a subset $E \subset \Omega$
of measure smaller than $\varepsilon$ such that
$f(u\_n) \to f(u)$ uniformly on $\Omega \setminus E$.


On $\Omega \setminus E$ we can use
\begin{equation\*}
 \int\_{\Omega \setminus E} f(u\_n) \, |\nabla v\_n|^2 \, \mathrm{d}x
 =
 \int\_{\Omega \setminus E} f(u) \, |\nabla v\_n|^2 \, \mathrm{d}x.
 +
 \int\_{\Omega \setminus E} (f(u\_n) - f(u)) \, |\nabla v\_n|^2 \, \mathrm{d}x
\end{equation\*}
For the first addend, we can use weak lower semicontinuity,
whereas the second addend goes to zero.
On $E$ we use the simple estimate
\begin{equation\*}
 \int\_{E} f(u\_n) \, |\nabla v\_n|^2 \, \mathrm{d}x \ge 0.
\end{equation\*}
Alltogether, we get
\begin{equation\*}
 \liminf\_{n \to \infty} J(u\_n, v\_n)
 \ge
 \int\_{\Omega \setminus E} f(u) \, |\nabla v|^2 \, \mathrm{d}x
 =
 J(u,v)
 -
 \int\_{E} f(u) \, |\nabla v|^2 \, \mathrm{d}x
 .
\end{equation\*}
Since $f(u) \, |\nabla v|^2 \in L^1(\Omega)$, we have
\begin{equation\*}
 \int\_{E} f(u) \, |\nabla v|^2 \, \mathrm{d}x
 \to
 0
\end{equation\*}
for $\varepsilon \searrow 0$.
This shows
\begin{equation\*}
 \liminf\_{n \to \infty} J(u\_n, v\_n)
 \ge
 J(u,v)
 .
\end{equation\*}


