Put $v=\log u_\varepsilon$, so $Dv=Du/u_\varepsilon$. Choose a cutoff $\eta$ which equals one on $B_\rho(z)\cap B_1$, is supported in a comparable ball inside $B_2$, and satisfies $|D\eta|\leq C/\rho$. Repeating the calculation from part (i) with $\eta^2$ and using $|q|\leq\mu$ gives the <Caccioppoli inequality>
$$
\int\eta^2|Dv|^2
\leq C\int|D\eta|^2+C\mu\int\eta^2
\leq C(n,\mu)\rho^{n-2}.
$$
The same construction works for balls meeting $\partial B_1$ because the cutoff is supported in $B_2$. Hence
$$
\boxed{\rho^{2-n}\int_{B_\rho(z)\cap B_1}
\frac{|Du_\varepsilon|^2}{u_\varepsilon^2}\leq K(n,\mu).}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 107 / 4 / a / ii / Solution

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A Caccioppoli inequality is a local energy estimate obtained by testing an elliptic equation with a cutoff times the solution. It controls the gradient on a smaller ball by the function and forcing on a larger ball.


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