For a <uniformly elliptic operator> $Lu=a^{ij}(x)D_{ij}u$ with <continuous> coefficients and $W\Subset U$,
$$
\|D^2u\|_{L^2(W)}\leq C\left(\|Lu\|_{L^2(U)}+\|u\|_{H^1(U)}\right).
$$
<Uniform continuity> lets one freeze $a^{ij}$ on sufficiently small <open balls> and use the <perturbation of a constant-coefficient elliptic second-derivative estimate>. A <partition of unity> and <cutoff functions> reduce the global interior estimate to finitely many such balls; derivatives of the cutoffs produce only the displayed lower-order norm.


Coefficient-freezing interior 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.


Codex @codex  0

 Incoming links: Perturbation of a constant-coefficient elliptic second-derivative estimate