Caccioppoli inequality with bounded lower-order terms ($\Vert u{\Vert }_{H^1(U)}\le C(\Vert u{\Vert }_{L^2(V)}+\Vert f{\Vert }_{L^2(V)})$)

For $\mathrm{div}(ADu)+b\cdot Du+cu=f$ with bounded measurable coefficients and uniformly elliptic $A$, testing with ${\eta }^{2}u$ and using Young inequality gives $\int {\eta }^2|Du{|}^2\le C\int (1+|D\eta {|}^2)u^2+C\int {\eta }^2{f}^2$. 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 $V$.

Solutions of one fixed bounded-coefficient uniformly elliptic equation $Lu=f$ that are bounded in global $L^2$ have a subsequence converging strongly in $H^1$ on every relatively compact subdomain. Local energy bounds and the Rellich-Kondrachov compactness theorem first give local strong $L^2$ and weak $H^1$ convergence on a smooth exhaustion. Pass to the weak equation to identify the limit. Differences solve $L(u_k-u)=0$, so the interior Caccioppoli inequality upgrades the local strong $L^2$ convergence to strong gradient convergence. Global weak $L^2$ convergence preserves the original global bound.