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?