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 $b_iD_i u+cu$ are controlled by the <Hölder interpolation inequality>
$$
[Du]_{\alpha;B_r}+|u|_{\alpha;B_r}
\leq\eta[D^2u]_{\alpha;B}+C_\eta|u|_{2;B}.
$$
Choosing $\eta$ in terms of the prescribed $\delta$ yields
$$
\boxed{[D^2u]_{\alpha;B_{1/2}}
\leq\delta[D^2u]_{\alpha;B}+C(n,\alpha,\beta,\delta)|u|_{2;B}.}
$$

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, producing
$$
\boxed{|u|_{2,\alpha;B_{1/2}}
\leq C(n,\alpha,\beta)|u|_{0;B}.}
$$
Equivalently, a contradiction-and-rescaling proof would produce a globally Hölder harmonic limit forbidden by the <Polynomial-growth Liouville theorem for harmonic functions>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-107/2/a/solution