Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following regularity estimates hold for $p=1$?

$$||u||_{W^{k+1,p}(\Omega)}\leq C(||f||_{W^{k,p}(\Omega)}+||u||_{L^p(\Omega)}),$$

for some constant independent of $u$? This is true for $p>1$, but I am not finding any reference for $p=1$. Any hint would be appreciated.