The <De Giorgi-Nash-Moser theorem> gives numbers $\alpha=\alpha(n,\lambda,\Lambda)\in(0,1)$ and $C=C(n,\lambda,\Lambda,\theta)$ such that every weak solution satisfies
$$
\boxed{\|u\|_{C^{0,\alpha}(B_\theta)}
\leq C\|u\|_{L^2(B_1)}}
$$
for each fixed $0<\theta<1$. In particular, every such weak solution has a locally Hölder-continuous representative despite the coefficients being only bounded and measurable.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 3 / a / Solution

If $|Du|\leq L$, the eigenvalues of $a^{ij}(Du)$ lie between $(1+L^2)^{-3/2}$ and $1$, so the differentiated equations are uniformly elliptic with bounded measurable coefficients depending only on $L$. Apply the <De Giorgi-Nash-Moser theorem> to each $D_ku$. Since $|D_ku|\leq L$, for every $\theta<1$,
$$
[Du]_{C^{0,\alpha}(B_\theta)}\leq C(n,L,\theta)
$$
for some $\alpha=\alpha(n,L)\in(0,1)$. The map
$$
p\longmapsto\delta_{ij}-\frac{p_ip_j}{1+|p|^2}
$$
is smooth with bounded derivative on $|p|\leq L$. Composition therefore gives
$$
\boxed{\|b^{ij}\|_{C^{0,\alpha}(B_\theta)}\leq\beta(n,L,\theta).}
$$

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 3 / b / ii / Solution

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{wiki}

If $u\in W^{1,2}(B_1)$ weakly solves $D_i(a^{ij}D_ju)=0$ with bounded measurable uniformly elliptic coefficients, then $u$ is locally Hölder continuous. For $0<\theta<1$, some $\alpha\in(0,1)$ and $C$ depending only on the dimension, ellipticity bounds, and $\theta$ satisfy
$$
\|u\|_{C^{0,\alpha}(B_\theta)}\leq C\|u\|_{L^2(B_1)}.
$$

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