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For a uniformly elliptic operator $L=a^{ij}(x)D_{ij}$ with continuous coefficients, one precise regularity statement concerns its graph norm closure. Suppose smooth $u_k$ satisfy $u_k\to u$ in $H^1(U)$ and $L{u}_k\to f$ in $L^2(U)$. On every $W⋐U$, apply the coefficient-freezing interior second-derivative estimate to $u_k-u_l$:The right side tends to zero, so $(u_k)$ is Cauchy in $H^2(W)$. Its limit is $u$, giving $u\in H_{\mathrm{loc}}^2(U)$. Since the coefficients are locally bounded, $a^{ij}{D}_{ij}{u}_k\to a^{ij}{D}_{ij}u$ locally in $L^2$, and hence $Lu=f$.

This proof requires the stated graph approximation. It does not assign an automatic distributional meaning to $a^{ij}{D}_{ij}u$ for every $u\in H^1$ when $a^{ij}$ is merely continuous: multiplication of an arbitrary distribution by a nonsmooth coefficient is not generally defined. A different weak or generalized solution class needs its own definition and approximation theorem; the divergence-form variational definition cannot simply be transferred to this expression.

The continuous coefficients are uniformly continuous on a compact neighborhood of $\overline{W}$. For every $x_0\in \overline{W}$, choose a ball $B_r(x_0)⋐U$ small enough thatfor all $i,j$. Part 4(b), with the frozen symmetric matrix $A=(a^{ij}(x_0))$, then applies on this ball.

Choose a finite collection of these balls and a smooth partition of unity $({\eta }_k)$ that sums to one near $\overline{W}$, with each ${\eta }_k$ supported in its corresponding ball. Applying part 4(b) to ${\eta }_{k}u$ givesSince $(a^{ij})$ is symmetric, the Leibniz rule gives the commutator formulaThe coefficients and the finitely many derivatives of the cutoff functions are bounded, soFinally $u={\sum }_k{\eta }_{k}u$ near $\overline{W}$. Summing the finite set of local estimates provesThis is the coefficient-freezing interior second-derivative estimate.