Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 2 / a

Freeze the lower-order coefficients on a ball and apply the interior Schauder estimate for the Poisson equation to a cutoff of $u$. The terms $b_i{D}_{i}u+cu$ are controlled by the Hölder interpolation inequalityChoosing $\eta$ in terms of the prescribed $\delta$ yields

The standard iteration lemma for nested balls absorbs the first term. The remaining $C^2$ norm is bounded by the interior derivative estimate and interpolation, producingEquivalently, a contradiction-and-rescaling proof would produce a globally Hölder harmonic limit forbidden by the Polynomial-growth Liouville theorem for harmonic functions.