Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 107 / 3 / b / ii / Solution

Differentiate the equation for $v_R$ with respect to $x_k$. The derivative $w=D_k{v}_R$ is a weak solution ofwhereThe eigenvalue in the direction of $p$ is $(1+|p{|}^2{)}^{-3/2}$ and every orthogonal eigenvalue is $(1+|p{|}^2{)}^{-1/2}$. Thus a uniform bound on $|Dv|$ makes this a uniformly elliptic operator with constants independent of $R$.