Let $ \Omega\subset\mathbb{R}^d $ be a Lipschitz domain and $ A=(a_{ij}(y)):\Omega\to\mathbb{R}^{d\times d} $ be a matrix valued function with uniformly elliptic conditions i.e. $ a_{ij}(y)\xi_i\xi_j\geq\lambda|\xi|^2 $ with $ \lambda>0 $. Consider the equation $$ -\operatorname{div}(A\nabla u)=F+\operatorname{div}(G) $$ where $ F\in L^p(\Omega) $ and $ G\in L^q(\Omega) $ with $ p,q $ to be determined. For $ \psi\in C_0^{\infty}(\Omega) $, we can choose test function $ \psi^2 u $ and get $$ \int_{\Omega}|\nabla u|^2|\psi|^2dx\leq C\left\{\int_{\Omega}|u|^2|\nabla\psi|^2dx+\int_{\Omega}|G|^2|\psi|^2dx+\int_{\Omega}|F||u||\psi|^2dx\right\}. $$ If $ B\subset 2B\subset\Omega $ with $ B(x,r) $, we can choose $ \psi $ be the cut off function $ \phi\in C_{0}^{\infty}(2B) $ with $ \phi\equiv 1 $ in $ B $ and $ |\nabla\phi|\leq\frac{C}{r} $. When $ d\geq 3 $, we can get that for $ q=2d/(d+2) $, \begin{eqnarray} \int_{\Omega}\phi^2|F||u|dx&\leq&\left(\int_{\Omega}(\phi|u|)^{2d/(d-2)}dx\right)^{(d-2)/(2d)}\left(\int_{\Omega}(\phi|F|)^qdx\right)^{1/q}\\ &\leq&\left(\int_{\Omega}|\nabla(\phi u)|^2dx\right)^{1/2}\left(\int_{\Omega}(\phi|F|)^qdx\right)^{1/q}\\ &\leq&\frac{1}{4}\int_{\Omega}|\nabla u|^2\phi^2 dx+\frac{1}{4}\int_{\Omega}|\nabla\phi|^2|u|^2dx+C\left(\int_{\Omega}|\phi F|^qdx\right)^{2/q}. \end{eqnarray} Then we have $$ \left(\int_{B}|\nabla u|^2dx\right)^{1/2}\leq\frac{C}{r}\left(\int_{2B}|u|^2dx\right)^{1/2}+C\left(\int_{2B}|f|^2dx\right)^{1/2}+r\left(\int_{2B}|F|^qdx\right)^{1/q},$$ where $ C $ is a constant independent of $ u $.
I want to ask can we generalize the Caccioppoli inequality to the dimension $ d=2 $? As the Sobolev index $ 2d/(d-2) $ may fail, I can not go on. Can you give me some hints or references?