= Coefficient-freezing interior second-derivative estimate
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.
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