Mollification of a divergence-forcing equation ($\Delta u_{\sigma }=\mathrm{div}(bu{)}_{\sigma }+(cu{)}_{\sigma }+f_{\sigma }$)

The distributional equation $\Delta u=\mathrm{div}(bu)+cu+f$ becomes the smooth interior equation $\Delta u_{\sigma }=\mathrm{div}(bu{)}_{\sigma }+(cu{)}_{\sigma }+f_{\sigma }$. The products are convolved as wholes. Only coefficient bounds on compact interior sets are needed, and the divergence-forcing interior H1 estimate provides a uniform first derivative bound.