Differentiate the equation for $v_R$ with respect to $x_k$. The derivative $w=D_kv_R$ is a weak solution of
$$
D_i\bigl(a^{ij}(Dv(Rx))D_jw\bigr)=0,
$$
where
$$
a^{ij}(p)=\frac{\delta^{ij}}{\sqrt{1+|p|^2}}
-\frac{p_ip_j}{(1+|p|^2)^{3/2}}.
$$
The 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 $R$.

Solution

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

 Solution

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