= Caccioppoli inequality with bounded lower-order terms
{c}
{title2=$\|u\|_{H^1(U)}\leq C(\|u\|_{L^2(V)}+\|f\|_{L^2(V)})$}
For $\operatorname{div}(A Du)+b\cdot Du+cu=f$ with bounded measurable coefficients and uniformly elliptic $A$, testing with $\eta^2u$ and using <Young inequality> gives $\int\eta^2|Du|^2\leq C\int(1+|D\eta|^2)u^2+C\int\eta^2f^2$. No sign condition on $c$ is needed for this interior estimate. A cutoff equal to one on $U$ and supported in $V$ yields the displayed bound; the constant depends only on dimension, ellipticity, coefficient bounds and the separation of $U$ from the boundary of $V$.
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