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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.

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.