Let $\Omega \subset \textbf{R}^N$ be open and bounded. Let $f \in L^2(\Omega)$ and let $u \in H^1(\Omega)$ be a weak solution of the equation \begin{align} Lu \equiv - \sum_{i,j=1}^n D_i(a_{ij}D_ju) = f. \tag{1} \end{align} This is defined by \begin{align} \int_{\Omega} \sum_{i,j=1}^n a_{ij}D_i u D_j v ~dx = \int_{\Omega}fv~dx \tag{2} \end{align} for all $v \in H_0^1(\Omega)$. Suppose that there is a constant $\Theta > 0$ such that for all $x \in \Omega$ \begin{align} \sqrt{ \sum_{i,j = 1}^n (a_{ij}(x))^2 } \leq \Theta. \tag{3} \end{align} By the Cauchy-Schwarz inequality in $\textbf{R}^n$ this also implies that \begin{equation} \sum_{i,j=1}^n a_{ij}(x) \zeta_i \xi_j \leq \Theta |\zeta| |\xi| \tag{4} \end{equation} for all $x \in \Omega$ and all $\zeta, \xi$ in $\textbf{R}^n$. Let us furthermore assume that there is a constant $\theta > 0$ s.t. \begin{align} \sum_{i,j = 1}^n a_{ij}(x)\xi_i \xi_j \geq \theta |\xi|^2 \tag{5} \end{align} for all $x \in \Omega$ and $\xi \in \textbf{R}^n$.
Let $\eta \in C^1_c(\Omega)$ be arbitrary. Prove the following a priori estimate: There holds \begin{align} \int_{\Omega}|Du|^2\eta^2dx \leq C_0 \left ( \int_{\Omega}u^2(|D\eta|^2 + \eta^2)dx + \int_U f^2\eta^2 dx \right ) \tag{*} \end{align} where $C_0$ depends only on $\theta$ and $\Theta$.
The first step here is to use $v = u \eta^2$ and plug it into the definition of the weak solution which is stated in equation $(2)$. The next step would be to use Cauchy-Schwarz and Peter-Paul together with the above conditions on $(a_{ij})$, but this is the point where im currently stuck. I would be really happy if someone could help me out!
Cheers, Pinch