Strong local Sobolev compactness for a fixed elliptic equation ($u_k⟶u\text{ in }{H}_{\mathrm{loc}}^1$)

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.