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

Put $v=\log u_{\epsilon }$, so $Dv=Du/u_{\epsilon }$. 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 $|D\eta |\le C/\rho$. Repeating the calculation from part (i) with ${\eta }^2$ and using $|q|\le \mu$ gives the Caccioppoli inequalityThe same construction works for balls meeting $\partial B_1$ because the cutoff is supported in $B_2$. Hence