I have the following statement:
Consider $F(u) = \int_a^b f(x,u(x))\, dx$ with $f \in C^1(a,b)$ and such that $f$ is convex in $u$. Also we further assume $f \ge 0$ for any $x,u$.
If $u_n \to u$ weakly in $L^1$, then $\liminf_{n \to +\infty} F(u_n) \ge F(u)$.
It resambles Tonelli's theorem, my try to give a proof is the following:
Since $f$ is convex in $u$ we have that $$F(u_n) = \int_a^b f(x,u_n(x))\, dx \ge \int_a^b f(x,u(x)) \, dx + \int_a^b f_u(x,u(x))(u_n - u) \, dx$$ Noting that $f_u$ is continous over a compact, follows that $f_u \in L^\infty$. Therefore $\int_a^b f_u(x,u(x))(u_n - u)) \, dx \to 0$ since $u_n \to u$ weakly in $L^1$. Therefore by taking $\liminf$ on both sides we conclude the proof.
Is the proof correct? There are some missing details that I should take care of?