EDIT: I should clarify the last question. Dorian has given me a good reference regarding weak continuity/semicontinuity of functionals which involve derivatives and depend on the spatial variables. The question that remains is simpler: when you have a functional $\int f(x,u) dx$ that is, say, continuous with respect to weak-* convergence in $L^\infty$, is it necessarily the case that $f(x,\cdot)$ is affine in $u$ at almost every point $x$? This question is purely measure theoretic and has nothing to do with derivatives (so without loss of generality I am asking about the measure space $I = [0,1]$ since I am interested in the case without atoms). By using a sequence of the form $\chi_{E_k} u + (1 - \chi_{E_k}) v$ for suitable characteristic functions $E_k$, we know that $f(x, \theta u + (1- \theta) v) = \theta f(x, u) + (1-\theta) f(x,v)$ for all measurable functions $\theta : I \to [0,1]$ and all bounded, measurable $u, v$ on $I$. Probably, this is enough to imply $f(x, u)$ is affine in $u$ for almost every $x$. Similarly, lower semicontinuity should imply convexity at almost every point.
It's well known that a functional of the form $u \mapsto \int f(u) dx$ is continuous with respect to weak convergence (say weak* convergence in $L^\infty$) if and only if the function $f$ is affine. An even more interesting thing to study is functionals which involve derivatives $u \mapsto \int f(du) dx$. The function $f$ needs to be quasiconvex to be lower semicontinuous.
I'm interested in functionals which also depend on the $x$ variable, like $\int f(x, u, du) dx$. Can anyone tell me a good place to read about continuity and semicontinuity with respect to weak convergence for functionals of this form. For example, must the $f(x,u)$ in $\int f(x,u) dx$ be convex / affine in the $u$ for almost every point $x$?