I'm looking for the following result:



> 
> Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. The map
>  $$u \mapsto \int\_0^T \int\_{\Omega} f(u(t))$$
>  is lower semicontinuous for $u \in L^2(0,T;L^2)$ where $f:\mathbb R \to \mathbb R$ is convex.
> 
> 
> 


Does anyone know how to prove this, or a reference for this result? Thank you.


I managed to this for a subsequence (so if $u\_n \to u$, I showed for $\liminf\_{n\_j \to \infty}$) but could not show it for the full sequence.


I asked this on StackExchange ([link](https://math.stackexchange.com/questions/827788/lower-semicontinuity-of-a-bochner-integral-of-a-convex-function)) but got not reply. 



Lower semicontinuity of a Bochner integral of a convex function
Use the following definition for lower semi continuity. That is, $f$ is lower semi continuous at $x\_0$ is $\liminf\_{x\rightarrow x\_0} f(x) \geq f(x\_0)$. This is equivalent to


For all $\epsilon > 0$ there exists $\delta > 0$ so that $\epsilon \geq f(x\_0) - f(x)$ for all $x \in B\_\delta(x\_0)$


Fix $u \in L^2(0,T;L^2(\Omega))$ and pick $\epsilon > 0$. One has \begin{eqnarray} F(u) - F(v) &=& \int\_0^T \int\_\Omega \left[ f(u) - f(v) \right]\end{eqnarray} Use the fact that convex functions are Lipshitz to get that $$F(u) - F(v) \leq C \int\_0^T \int\_\Omega \left| u - v\right|$$ for some constant $C$. From this you can easily conclude what you want by using the fact that you are playing over a bounded domain and using Holder. 


