On a ball $B_{2r}(x_0)\Subset B$, both
$$
M-u,
\qquad u-m,
\quad
M=\sup_{B_{2r}}u,
\quad m=\inf_{B_{2r}}u,
$$
are nonnegative solutions and hence supersolutions. Combining the <Weak Harnack inequality> with the local boundedness estimate for subsolutions gives an <oscillation decay estimate>
$$
\operatorname{osc}_{B_r(x_0)}u
\leq\theta\operatorname{osc}_{B_{2r}(x_0)}u,
\qquad0<\theta<1,
$$
where $\theta$ depends only on $n,\lambda,\Lambda$. Iteration yields <Hölder continuity> with some exponent $\mu\in(0,1)$. The local $L^2$-to-$L^\infty$ estimate controls the initial oscillation and gives
$$
\boxed{|u|_{0,\mu;B_{1/8}}
\leq C\lVert u\rVert_{L^2(B)}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 4 / b / Solution

For a weak solution of a uniformly elliptic divergence-form equation, local boundedness and the <Weak Harnack inequality> imply
$$
\operatorname{osc}_{B_r}u\leq\theta\operatorname{osc}_{B_{2r}}u
$$
for a universal $0<\theta<1$. Iteration gives local <Hölder continuity>.

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