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Let $L$ be a second order, uniformly elliptic, non variational operator with coefficients which are bounded and measurable in ${\mathbb{R}}^d$ ($d\ge 3$) and continuous in ${\mathbb{R}}^d\setminus \left\{0\right\}$. Then, if $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a bounded domain, we prove that $L\left(W^{2,p}\left(\mathrm{\Omega }\right)\right)$ is dense in $L^p\left(\mathrm{\Omega }\right)$ for any $p\in \left(1,d/2\right]$.