= Strong local Sobolev compactness for a fixed elliptic equation
{title2=$u_k\longrightarrow u\text{ in }H^1_{\mathrm{loc}}$}
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.
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