Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 107 / 3 / i / Solution

Here $w$ is the harmonic replacement of $u$ in $B_r=B(x_0,r)$, so $v=u-w\in H_0^1(B_r)$. Subtract the weak equations and test with $v$:Because $B_r\subset B(0,1)$, $|x_1|\le 1$. The Sobolev inequality $\Vert v{\Vert }_6\le C\Vert \nabla v{\Vert }_2$ and the Holder inequality giveAfter division and squaring,