= Solution
Choose nested interior open sets $U\subset\subset V\subset\subset W\subset\subset\Omega$. For small $\sigma$, part (b) gives a smooth equation on $V$ of the form
$$
\Delta u_\sigma=\operatorname{div}F_\sigma+G_\sigma,\qquad F_\sigma=(bu)_\sigma,\quad G_\sigma=(cu)_\sigma+f_\sigma.
$$
The <divergence-forcing interior H1 estimate> is
$$
\|u_\sigma\|_{H^1(U)}\leq C\left(\|u_\sigma\|_{L^2(V)}+\|F_\sigma\|_{L^2(V)}+\|G_\sigma\|_{L^2(V)}\right).
$$
One can obtain this estimate directly by testing the smooth equation against $\eta^2u_\sigma$ and applying <Young inequality>, with a <cutoff function> equal to one on $U$ and supported in $V$. The <approximate identity> and the $L^2$ <convolution> bound give, uniformly in $\sigma$,
$$
\|u_\sigma\|_{L^2(V)}\leq\|u\|_{L^2(W)},\quad\|(bu)_\sigma\|_{L^2(V)}\leq\|b\|_{L^\infty(W)}\|u\|_{L^2(W)},
$$
with analogous bounds for $(cu)_\sigma$ and $f_\sigma$. The coefficients need only be bounded on $W$. Therefore $u_\sigma$ is bounded in $H^1(U)$. It converges to $u$ in $L^2(U)$, and <weak sequential compactness in a Hilbert space> in this <Sobolev space> gives $u\in H^1(U)$. Since $U$ was arbitrary, $u\in H^1_{\mathrm{loc}}(\Omega)$.
Now expand the <distributional derivative>:
$$
\Delta u=b\cdot\nabla u+(\operatorname{div}b+c)u+f.
$$
Its right side belongs to $L^2_{\mathrm{loc}}$, so the interior $H^2$ <elliptic regularity> estimate gives $u\in H^2_{\mathrm{loc}}$. More generally, if $u\in H^m_{\mathrm{loc}}$ for an integer $m\geq1$, multiplication by the smooth coefficients puts the right side in $H^{m-1}_{\mathrm{loc}}$. Interior <elliptic regularity> then gives $u\in H^{m+1}_{\mathrm{loc}}$. This <elliptic regularity bootstrap> proves $u\in H^k_{\mathrm{loc}}$ for every integer $k$.
For any nonnegative integer $\ell$, choose $k>\ell+n/2$. The <Sobolev embedding theorem> on compact interior subsets gives $u\in C^\ell_{\mathrm{loc}}$. Taking all $\ell$ proves \b[$u\in C^\infty(\Omega)$], for its smooth representative. The first gain from $L^2$ to $H^1$ is the step supplied by the mollified equation; assuming $H^1$ in the initial definition would miss that step.
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