Fix $W\Subset V\Subset U$. The standard local regularization of the maximal <graph norm> domain supplies smooth compactly supported approximants $u_k$ on $V$ such that
$$
u_k\longrightarrow u\quad\text{in }H^1(V),
\qquad
Lu_k\longrightarrow Lu\quad\text{in }L^2(V).
$$
Apply part 4(c), with $V$ as the outer domain, to $u_k-u_m$. It gives
$$
\|D^2(u_k-u_m)\|_{L^2(W)}
\leq C\left(\|L(u_k-u_m)\|_{L^2(V)}
+\|u_k-u_m\|_{H^1(V)}\right)\longrightarrow0.
$$
Thus $(u_k)$ is Cauchy in $H^2(W)$. Its $H^1$ limit is $u$, so $u\in H^2(W)$. Since $W\Subset U$ was arbitrary, the definition of a <Local Sobolev space> gives
$$
\boxed{u\in H^2_{\mathrm{loc}}(U).}
$$
This is the <Interior H2 regularity for continuous nondivergence coefficients>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 4 / d / Solution

{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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 Incoming links: Interior H2 regularity for continuous nondivergence coefficients