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

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.