I am interested in the following problem:
Is there any function (let's say continuous) $W:\mathbb{R}\to \mathbb{R}$ satisfying $W(x+1)=W(x) \forall x\in \mathbb{R}$ and such that the functional $L^2([0,1])\to \mathbb{R},\ f\mapsto \int_{0}^1 W(f(x))dx$, is weakly lower-semicontinuous? . Of course, $W=cons$ is possible but I am looking for other solutions.
There are not other functions. If $f \mapsto \int W(f) \ dx$ is weakly lower semicontinuous from $L^2$ to $\mathbb R$ then $W$ is convex. Due to the periodicity requirements, only constant functions qualify.
Assume $W$ is not convex. Then there are $v_1,v_2$ and $\lambda\in (0,1)$ with $W(\lambda v_1+ (1-\lambda v_2)) > \lambda W(v_1) + (1-\lambda)W(v_2)$.
We will now construct an oscillating sequence: Define $\phi(x):= \chi_{[0,\lambda)}v_1 + \chi_{[\lambda,1)}v_2$ and extend it $1$-periodically to a function from $\mathbb R$ to $\mathbb R$. Define $u_n(x) := \phi(nx)$, $x\in (0,1)$. Then $$ \int_0^1 W(u_n) = \lambda W(v_1) + (1-\lambda)W(v_2). $$ The sequence $(u_n)$ converges weakly in $L^2$ to the constant function $u(x)=\lambda v_1+(1-\lambda)v_2$. And $$ \int_0^1 W(u) = W(\lambda v_1+(1-\lambda)v_2)> \lambda W(v_1) + (1-\lambda)W(v_2) = \int_0^1 W(u_n). $$ And $u\mapsto \int W(u)$ is not weakly lower semicontinuous.