= Interior H2 regularity for continuous nondivergence coefficients
{c}
{title2=$Lu\in L^2(U)\Longrightarrow u\in H^2_{\mathrm{loc}}(U)$}
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 $Lu_k\to f$ in $L^2(U)$. On every $W\Subset U$, apply the <coefficient-freezing interior second-derivative estimate> to $u_k-u_l$:
$$
\|D^2(u_k-u_l)\|_{L^2(W)}\le C\bigl(\|Lu_k-Lu_l\|_{L^2(U)}+\|u_k-u_l\|_{H^1(U)}\bigr).
$$
The right side tends to zero, so $(u_k)$ is Cauchy in $H^2(W)$. Its limit is $u$, giving $u\in H^2_{\rm loc}(U)$. 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.
Back to article page
