Let $\Omega\subset \mathbb{R}^N$ (regular domain) and $u\in W^{1,\infty}(\Omega)$ such that $\| \nabla u \|_{L^\infty} \le 1$ (i.e., 1-Lipschitz function on $\Omega$). Let $p_n\overset{\ast}{\rightharpoonup} 0$ in $L^{\infty}(\Omega)^N$, i.e. $\int_\Omega p_n f dx \to 0$ for every $f$ in $L^1(\Omega)^N$.
I don't know if:
Does there exist a sequence $\{u_n\}\subset W^{1,\infty}(\Omega)$ satisfying $\|\nabla u_n -p_n\|_{L^\infty}\le 1$ and $u_n\overset{\ast}{\rightharpoonup}u$ in $L^\infty(\Omega)$ ?