We have \begin{equation} L u=\sum_{i, j=1}^n \partial_i\left(a_{ij} \partial_j u\right) \end{equation} and $u$ a solution of the weak Dirichlet problem $$ \left\{\begin{array}{l} L u=f \text { weakly in } \Omega, \\ u \in W_0^{1,2}(\Omega) . \end{array}\right. $$ I want to show $$ \|u\|_{W^{1,2}} \leq \rho\left(\operatorname{diam}(\Omega)+\operatorname{diam}(\Omega)^2\right)\|f\|_{L^2}, $$ where $\rho$ is the ellipticity constant of $L$.
Any hints?
Testing the equation with $u$ should give $$ \rho^{-1} \|\nabla u\|_{L^2}^2 \le \|f\|_{L^2} \|u\|_{L^2} \le \|f\|_{L^2} \cdot diam(\Omega) \|\nabla u\|_{L^2},$$ which is almost the desired inequality.