Let $L$ be a [[Second Order Elliptic Operator|second order elliptic operator]] in divergence form $ Lu = -\div{ADu} + \anglesn{b, Du}_{\real^d} + cu $ where $A \in C^1(U; \real^{d \times d})$, $b \in L^\infty(U; \real^d)$, $c \in L^\infty(U)$, and $f \in L^2(U)$. Let $u \in H^1(U)$ be a [[Weak Solution of Second Order Linear PDE|weak solution]] to the equation $Lu = f$, then $u \in H^2_{\text{loc}}(U)$ where for each $V \subset \subset U$, there exists $C_{L, V, U} \ge 0$ such that $ \norm{u}_{H^2(V)} \le C_{L, V, U}(\norm{f}_{L^2(U)} + \norm{u}_{L^2(U)}) $ *Proof*. Let $ \td f = f - \anglesn{b, Du}_{\real^d} - cu $ then since $u \in H^1(U)$ is a weak solution, for any $\phi \in H_0^1(U)$, $ \anglesn{ADu, D\phi}_{L^2(U; \real^d)} = \anglesn{\td f, \phi}_{L^2(U)} $ Let $W \subset \subset U$ such that $V \subset\subset W$, and $\eta \in C_c^\infty(U; [0, 1])$ with $\eta|_V = 1$, $\eta|_{U \setminus W} = 0$. Let $\delta > 0$ such that $V + he_\ell \subset W$ for all $1 \le \ell \le d$ and $h \in (-\delta, \delta)$. Fix $1 \le \ell \le d$, $h \in (-\delta, \delta)$, and take $\phi = -D_\ell^{-h}(\eta^2D_\ell^h u)$. Using integration by parts and the product rule for difference quotients, $ \begin{align*} \anglesn{ADu, D\phi}_{L^2(U; \real^d)} &= -\anglesn{ADu, D(D_\ell^{-h}(\eta^2D_\ell^hu))}_{L^2(U; \real^d)} \\ &= -\anglesn{ADu, D_\ell^{-h}(D(\eta^2D_\ell^hu))}_{L^2(U; \real^d)} \\ &= \anglesn{D_\ell^h(ADu), D(\eta^2D_\ell^hu)}_{L^2(U; \real^d)} \\ &= \anglesn{(\tau_{-he_j}A)D_\ell^h(Du), D(\eta^2D_\ell^hu)}_{L^2(U; \real^d)} \\ &+ \anglesn{(D_\ell^hA)Du, D(\eta^2D_\ell^hu)}_{L^2(U; \real^d)} \end{align*} $ By further expanding $ \begin{align*} D(\eta^2D_\ell^hu) &= D\eta^2 \cdot D_\ell^hu + \eta^2D_\ell^hDu \\ &= 2\eta \cdot D\eta \cdot D_\ell^hu + \eta^2D_\ell^hDu \end{align*} $ This allows decomposing $\anglesn{ADu, D\phi}_{L^2(U; \real^d)} = S + R$, where $ \begin{align*} S &= \anglesn{(\tau_{-he_j}A)D_\ell^h(Du), \eta^2D_\ell^hDu}_{L^2(U; \real^d)} \\ R &= 2\anglesn{(\tau_{-he_j}A)D_\ell^h(Du), \eta \cdot D\eta \cdot D_\ell^hu}_{L^2(U; \real^d)}\\ &+ \anglesn{(D_\ell^hA)Du, \eta^2D_\ell^hDu }_{L^2(U; \real^d)} \\ &+ \anglesn{(D_\ell^hA)Du,\eta \cdot D\eta \cdot D_\ell^hu}_{L^2(U; \real^d)} \end{align*} $ Let $\theta > 0$ such that $A(x)(\xi) \ge \theta\abs{\xi}$ for all $\xi \in \real^d$ and $x \in U$, then since $\eta|_V = 1$, $S \ge \theta\normn{\eta D_\ell^hDu}_{L^2(V; \real^d)}^2$. To bound $R$ from above, by assumption on $h$, $ \begin{align*} &|{2\anglesn{(\tau_{-he_j}A)D_\ell^h(Du), \eta \cdot D\eta \cdot D_\ell^hu}_{L^2(U; \real^d)}}| \\ &\le 2\norm{A}_{L^\infty(W; \real^{d \times d})}\norm{D\eta}_{L^\infty(W; \real^d)} \cdot \normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}\normn{D_\ell^hu}_{L^2(W)} \\ &\le 4\norm{A}_{L^\infty(W; \real^{d \times d})}\norm{D\eta}_{L^\infty(W; \real^d)} \cdot \normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}\normn{Du}_{L^2(U; \real^d)} \\ &\le C_{A, \eta} \normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}\normn{Du}_{L^2(U; \real^d)} \end{align*} $ For the last two terms, since $A \in C^1(U; \real^{d \times d})$ and $W \subset \subset U$, $\norm{DA}_{L^\infty(W; \real^{(d \times d) \times d})} < \infty$. Thus $ \begin{align*} &|{\anglesn{(D_\ell^hA)Du, \eta^2D_\ell^hDu }_{L^2(U; \real^d)}} + {\anglesn{(D_\ell^hA)Du,\eta \cdot D\eta \cdot D_\ell^hu}_{L^2(U; \real^d)}}| \\ &\le \norm{DA}_{L^\infty(W; \real^{(d \times d) \times d})}\norm{Du}_{L^2(W; \real^{d})}{\normn{\eta D_\ell^hDu}_{L^2(W; \real^{d \times d})}} \\ &+ \norm{DA}_{L^\infty(W; \real^{(d \times d) \times d})}\norm{Du}_{L^2(W; \real^d)}\norm{D\eta}_{L^\infty(W; \real^d)}\normn{D^h_\ell u}_{L^2(W)}\\ &\le \norm{DA}_{L^\infty(W; \real^{(d \times d) \times d})}\norm{Du}_{L^2(W; \real^{d})}{\normn{\eta D_\ell^hDu}_{L^2(W; \real^{d \times d})}} \\ &+ 2\norm{DA}_{L^\infty(W; \real^{(d \times d) \times d})}\norm{D\eta}_{L^\infty(W; \real^d)}\norm{Du}_{L^2(U; \real^d)}^2 \\ &\le C_{A, \eta}({\norm{Du}_{L^2(W; \real^d)}{\normn{\eta D_\ell^hDu}_{L^2(W; \real^{d \times d})}} + \norm{Du}_{L^2(U; \real^d)}^2}) \end{align*} $ Combing these, $ \abs{R} \le C_{A, \eta}\normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}\norm{Du}_{L^2(W; \real^d)} +C_{A, \eta}\norm{Du}_{L^2(U; \real^d)}^2 $ By [[Young's Inequality]] with $\eps = \theta/2$, $ \abs{R} \le \frac{\theta}{2}\normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}^2 + \paren{\frac{C_{A, \eta}^2}{2\theta} + C_{A, \eta}}\norm{Du}_{L^2(W; \real^d)}^2 $ Thus $ \abs{\anglesn{ADu, D\phi}_{L^2(U; \real^d)}} \ge \frac{\theta}{2}\normn{\eta D_\ell^hDu}_{L^2(W; \real^d)} - C_{A, \eta, \theta}\norm{Du}_{L^2(W; \real^d)}^2 $ On the flip side, $ \begin{align*} \abs{\anglesn{\td f, \phi}_{L^2(U)}} &\le \braks{\norm{f}_{L^2(U)} + \norm{b}_{L^\infty(U; \real^d)}\norm{Du}_{L^2(U; \real^d)} + \norm{c}_{L^\infty(U)}\norm{u}_{L^2(U)}}\norm{\phi}_{L^2(U)} \\ &\le C_{b, c}\braks{\norm{f}_{L^2(U)} + \norm{Du}_{L^2(U; \real^d)} + \norm{u}_{L^2(U)}}\norm{\phi}_{L^2(U)} \end{align*} $ where $ \begin{align*} \norm{\phi}_{L^2(U)} &\le 2\normn{D(\eta^2D_\ell^hu)}_{L^2(U; \real^d)} \\ &\le 4\norm{D\eta}_{L^\infty(U; \real^d)}\normn{D_\ell^hu}_{L^2(W;\real^d)} + \normn{D_\ell^hDu}_{L^2(W; \real^d)} \\ &\le C_{\eta}(\norm{Du}_{L^2(U; \real^d)} + \normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}) \end{align*} $ By Young's inequality with $\eps = \theta/4$, $ \begin{align*} \abs{\anglesn{\td f, \phi}_{L^2(U)}} &\le C_{b, c, \eta}\norm{Du}_{L^2(U; \real^d)}\braks{\norm{f}_{L^2(U)} + \norm{Du}_{L^2(U; \real^d)} + \norm{u}_{L^2(U)}} \\ &+ \frac{C_{b, c, \eta}^2}{\theta}\braks{\norm{f}_{L^2(U)} + \norm{Du}_{L^2(U; \real^d)} + \norm{u}_{L^2(U)}}^2\\ &+ \frac{\theta}{4}\normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}^2 \\ &\le C_{b, c, \eta, \theta}\braks{\norm{f}_{L^2(U)}^2 + \norm{Du}_{L^2(U; \real^d)}^2 + \norm{u}_{L^2(U)}^2}+ \frac{\theta}{4}\normn{\eta D_\ell^hDu}_{L^2(W; \real^d)}^2 \end{align*} $ Combining with the estimate on $\anglesn{ADu, D\phi}_{L^2(U; \real^d)}$ yields that $ \normn{D_\ell^hDu}_{L^2(W; \real^d)}^2 \le C_{A, b, c, \eta, \theta}\braks{\norm{f}_{L^2(U)}^2 + \norm{Du}_{L^2(U; \real^d)}^2 + \norm{u}_{L^2(U)}^2} $