Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 2 / d / Solution

Let $u\in H_{\mathrm{loc}}^1(\Omega )$ weakly solvewhere $a^{ij},F^i\in C^{0,\alpha }$, $g\in L^p$ for a suitable $p>n/(1-\alpha )$, and $a^{ij}$ is uniformly elliptic. The interior divergence-form Schauder estimate states that for ${\Omega }^{\prime }⋐{\Omega }_1⋐\Omega$,For the homogeneous equation the final two terms vanish.