Compactness proof of a nonnegative elliptic solution estimate (${\sup }_{\Omega }u\le C({\inf }_{{\Omega }^{\prime }}u+\Vert Lu{\Vert }_{C^{0,\mu }})$)

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 ${\Omega }^{\prime }$ 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.