= Elliptic regularity bootstrap with smooth lower-order coefficients
{title2=$H^m_{\mathrm{loc}}\longrightarrow H^{m+1}_{\mathrm{loc}}$}
= Elliptic regularity bootstrap
{synonym}
Once an $L^2$ distributional solution of $\Delta u=\operatorname{div}(bu)+cu+f$ gains $H^1$ regularity, expand the divergence term. Smooth multiplication makes the right side belong to $H^{m-1}_{\mathrm{loc}}$ whenever $u\in H^m_{\mathrm{loc}}$. Interior Poisson regularity gives $H^{m+1}_{\mathrm{loc}}$. Iteration and the <Sobolev embedding theorem> produce a smooth representative.
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