Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type $F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even $\partial\Omega\in C^\infty$, $\Omega\subset\mathbb{R}^n$. I should emphasize that I'am not interested in weak sequentially lower semicontinuity (wSlsc), for which there are many good results, most of them concerned with quasiconvexity.

What can be said if the functional is only defined on a subset $Y\subset W^{1,2}(\Omega, \mathbb{R}^N)$?

I already know the following conditions

If the (topological) subspace $Y$ is metrizable, for example if $Y$ is a finite dimensional linear subspace or $Y'$ is separable and $Y$ is bounded in the norm, wlsc and wSlsc are the same, so that one could use the results mentioned above. Are there other, more general but still good to verify (i.e. more concrete than the Urysohn's metrization theorem), conditions, which assure that $Y$ is metrizable?

For a locally convex, Hausdorff, topological vector space and a convex functional we have wlsc $\Leftrightarrow$ strong lower semi continuity.