Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f\_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f\_n \to f$, $f\_n(0)=f(0)=0$ and $f\_n$ Lipschitz with Lipschitz constant depending on $n$. In fact $f\_n(x) := \int\_0^x |T\_n((|s|-\frac 1n)^+ + \frac 1n)|^{-\frac{1}{2}}$ where $T\_n(x) = x$ for $|x| \leq n$ and $T\_n(x) = n$ otherwise (the usual truncation function).


I have the following convergence results:
$$e\_n \to e \quad\text{in $X$}$$
$$\nabla e\_n \rightharpoonup \nabla e\quad\text{in $X$}$$
$$\nabla f\_n(e\_n) \rightharpoonup f^\*\quad\text{in $X$}$$
$$f\_n(e\_n) \to f(e) \quad\text{pointwise a.e.}$$
**I wish to idenfify $f^\*$ with $\nabla f(e)$.**


I also have additional uniform bounds on $f\_n(e\_n)$ and $e\_n$ in the space $L^\infty(0,T;L^\infty(\Omega))$. Unfortunately since the domain is unbounded we can't say anything about $f\_n(e\_n)$ being bounded in $L^2$.


A DCT argument doesn't work either. 


If it helps, 


Does anyone have any ideas or techniques to do this? 



Identifying the weak limit of a gradient (Bochner spaces)
Your uniform bounds in $L^{\infty}\_t L^{\infty}\_x$ will be of great help here. First, let us choose some big radius $R > 0$ and restrict our attention to the ball $B(0,R)$ instead of $\Omega$.


**UPDATE** : Here is a second attempt of a proof, with the same idea as before.


Let $\varphi$ be a function in $\mathcal{D}(]0,T[ \times \Omega)$ and choose $R$ big enough so as to cover the spatial support of $\varphi$.


From the weak convergence in $L^2\_t L^2\_x$, we know that 


$$< \nabla f\_n(e\_n), \varphi >\_{\mathcal{D}', \mathcal{D}} = \int \nabla f\_n(e\_n) \varphi \to \int f^\* \varphi .$$


On the other hand, because you have a uniform bound for $f\_n(e\_n)$ in $L^{\infty}\_t L^{\infty}\_x$ and that constants are integrable on $]0,T[ \times B(0,R)$, DCT tells you that 


$$\int f\_n(e\_n) \nabla \varphi \to \int f(e) \nabla \varphi . $$


The last term is equal to $$- < \nabla f(e), \varphi >\_{\mathcal{D}', \mathcal{D}}$$


and we conclude that $f^\*$ and $\nabla f(e)$ agree as distributions. As they are both functions, they also agree as functions, in $L^2\_t L^2\_x$ for instance.


Sorry again for the failed attempt, hope this one will be clearer.


(Notice one thing : you only need uniform bounds on $f\_n(e\_n)$ locally in space and time, not globally.)


