First replace $u$ by $u+\varepsilon$ and later let $\varepsilon\downarrow0$. In the supersolution inequality use the nonnegative test function
$$
\eta^2/(u+\varepsilon),
$$
where $\eta$ is compactly supported. Ellipticity and <Young inequality> give the <Logarithmic Caccioppoli inequality>
$$
\int_B\eta^2|D\log(u+\varepsilon)|^2
\leq C(n,\lambda,\Lambda)\int_B|D\eta|^2.
$$
Choose $\eta=1$ on $B_\rho$, supported in $B$, with $|D\eta|\leq C(1-\rho)^{-1}$. Letting $\varepsilon\downarrow0$ and using <Fatou lemma> gives
$$
\boxed{
\int_{B_\rho}\frac{|Du|^2}{u^2}
\leq C(1-\rho)^{-2}.}
$$

Solution

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

{c}

For a positive weak supersolution $u$ of a uniformly elliptic divergence-form equation, testing with $\eta^2/u$ gives
$$
\int\eta^2|D\log u|^2\leq C\int|D\eta|^2.
$$
The constant depends only on the ellipticity bounds.


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