Interior H2 regularity for continuous nondivergence coefficients ($Lu\in L^2(U)⟹u\in H_{\mathrm{loc}}^2(U)$)

Let $L=a^{ij}(x)D_{ij}$ be uniformly elliptic with continuous coefficients. The coefficient-freezing interior second-derivative estimate extends from smooth functions to the graph norm closure of $L$ on $H^1(U)$: if $u_k\to u$ in $H^1(U)$ and $L{u}_k\to f$ in $L^2(U)$, then $(u_k)$ is Cauchy in $H^2(W)$ for every $W⋐U$. Hence $u\in H_{\mathrm{loc}}^2(U)$ and $Lu=f$. Equivalently, the standard local regularization argument gives the same conclusion for weak solutions.