= Compactness proof of a nonnegative elliptic solution estimate
{title2=$\sup_\Omega u\leq C(\inf_{\Omega'}u+\|Lu\|_{C^{0,\mu}})$}
For a fixed <uniformly elliptic operator> with $C^{0,\mu}$ coefficients on a bounded smooth <connected> domain, nonnegative solutions with zero <Dirichlet boundary condition> satisfy the displayed estimate when $\Omega'$ is nonempty and compactly contained in $\Omega$. If it failed, normalize the global supremum to one. The forcing and the interior infimum then tend to zero. The <global Schauder estimate> and <compact embedding of Hölder spaces> produce a nonzero homogeneous limit with an interior zero, contradicting the <strong minimum principle for elliptic operators>. An arbitrary bounded zeroth-order coefficient is allowed: replace $c$ by $\min(c,0)$ for the positivity argument. Keeping global boundary control prevents the normalized maximum from disappearing in the limit.
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