Weak convergence of the vector norm a vector field?

Question

Suppose $\Omega\subset\mathbb{R}^n$ and $\mathbf{v}^k:\Omega\rightarrow \mathbb{R}^m$ is a sequence of vector fields on $\Omega$ that converges weakly in $L^p(\Omega,\mathbb{R}^m)$ to $\mathbf{v}$. Does the vector (finite-dimensional) norm, i.e. $\left\lvert\mathbf{v}^k\right\rvert:\Omega\rightarrow\mathbb{R}$ converge weakly in $L^p(\Omega)$ to $\left\lvert\mathbf{v}\right\rvert$?

Answer 1

No, this is not true, even in the simplest case $m = 1$. You can take, e.g., $\Omega = (0,1)$ and $$ v_k(x) = \operatorname{sign}(\sin(2\pi k x)).$$