Perturbation of a constant-coefficient elliptic second-derivative estimate
= Perturbation of a constant-coefficient elliptic second-derivative estimate
If $\|a^{ij}-A^{ij}\|_\infty<\varepsilon$ for every $i,j$, then the <Cauchy-Schwarz inequality> gives
$$
\|(a^{ij}-A^{ij})D_{ij}u\|_2\leq n\varepsilon\|D^2u\|_2.
$$
Consequently $\varepsilon\leq\theta/(2n)$ preserves half of the estimate's lower bound.