weak convergence in $L^2$ and convergence of integral involving test-functions

Question

Let $\Omega$ be a bounded set of $\mathbb{R}^n$ and $(f_n)_n\subset L^2(\Omega)$ such that $f_n\rightharpoonup f\in L^2(\Omega)$ weakly in $L^2(\Omega)$. Then for any given test function $\phi\in C^\infty_c(\Omega)$, do we have the following convergent property: $$ \int_\Omega |f_n|\phi\,dx\to \int_\Omega |f|\phi \,dx,\quad \textrm{as $n\to \infty$.} $$

Answer 1

Consider $n = 1$, $\Omega = (0,1)$ and let $$u_n(x) = \begin{cases} 1 \qquad k2^{-n} \leq x < (k+1)2^{-n} \text{ for an even } k\\ -1 \qquad \text{otherwise} \end{cases}$$

Then you can check that $u_n \rightharpoonup 0$ weakly in $L^2(\Omega)$ but $|u_n(x)| = 1$ for every $x$ so $\int |u_n| \phi \not \to 0$ for $\phi$ such that $\int \phi \neq 0$.

To check that $u_n \rightharpoonup 0$, first note that $\|u_n\|_{L^2} = 1$ for each $n$ and so it is enough to check that $\langle u_n, \phi \rangle \to 0$ for each $\phi \in C^\infty(0,1)$.

To do this, for any Lipschitz continuous $\phi$ write $$|\langle u_n, \phi \rangle| \leq \sum_{0 \leq k < 2^n, k \text{ even}} \int_{k2^{-n}}^{(k+1)2^{-n}} |\phi(x) - \phi(x+ 2^{-n})| dx \lesssim \sum_{0 \leq k < 2^n, k \text{ even}} 2^{-2n} = 2^{-n}.$$