= Interior H2 regularity for continuous nondivergence coefficients
{c}
{title2=$Lu\in L^2(U)\Longrightarrow u\in H^2_{\mathrm{loc}}(U)$}
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 $Lu_k\to f$ in $L^2(U)$, then $(u_k)$ is Cauchy in $H^2(W)$ for every $W\Subset U$. Hence $u\in H^2_{\mathrm{loc}}(U)$ and $Lu=f$. Equivalently, the standard local regularization argument gives the same conclusion for <weak solutions>.
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