= Annular Caccioppoli inequality
{title2=$\int_{B_R}|\nabla u|^2\leq CR^{-2}\int_{B_{2R}\setminus B_R}|u-c|^2$}
= Caccioppoli inequality with an annular right side
{c}
{synonym}
For a <weak solution> of a <uniformly elliptic> homogeneous <divergence-form elliptic operator>, testing with $\eta^2(u-c)$ and choosing the <gradient> of the <cutoff function> in $B_{2R}\setminus B_R$ gives $\int_{B_R}|\nabla u|^2\leq CR^{-2}\int_{B_{2R}\setminus B_R}|u-c|^2$. The arbitrary constant $c$ can be chosen as an annular average, which makes the estimate suitable for a <hole-filling argument>.
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