Perturbation of a constant-coefficient elliptic second-derivative estimate

If $\Vert a^{ij}-A^{ij}{\Vert }_{\infty }<\epsilon$ for every $i,j$, then the Cauchy-Schwarz inequality givesConsequently $\epsilon \le \theta /(2n)$ preserves half of the estimate's lower bound.

For a uniformly elliptic operator $Lu=a^{ij}(x)D_{ij}u$ with continuous coefficients and $W⋐U$,Uniform continuity lets one freeze $a^{ij}$ on sufficiently small open balls and use the perturbation of a constant-coefficient elliptic second-derivative estimate. A partition of unity and cutoff functions reduce the global interior estimate to finitely many such balls; derivatives of the cutoffs produce only the displayed lower-order norm.

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.