= Compactness of zero-boundary Poisson solutions
On a fixed smooth bounded domain, zero-boundary solutions with forcing in a bounded subset of $C^{0,\mu}$ are uniformly bounded in $C^{2,\mu}$ by a maximum-principle barrier and the <global Schauder estimate>. The <Arzelà-Ascoli theorem> gives convergence in $C^2$. A closed Hölder-norm constraint passes to the limit by <lower semicontinuity of a Hölder seminorm>, so the constrained solution set is compact rather than merely <relatively compact>.
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