So I am reading Theorem 1 in section 6.3.1 of PDE by L.C. Evans and there is a part in which I don't understand in the last part of the proof.
Let us define a new cutoff function $\xi := \begin{cases}\xi \equiv 1 \text{ on } W, \,spt\xi \subset U \\ 0 \leq \xi \leq 1\end{cases}$ given $V \subset\subset W \subset\subset U$ and $spt\xi$ is the support of $\xi$. Furthermore, we have the following identity :
$$\sum_{i,j=1}^{n}\int_{U}a^{ij}u_{x_{i}}v_{x_{j}}dx = \int_{U}(f- \sum_{i-1}^{n}b^{i}u_{x_{i}}-cu)vdx$$
given $a^{ij} \in C^{1}(U)$, $b^{i},c\in L^{\infty}(U)$ and $f \in L^{2}(U)$ with $u \in H^{1}_{loc}(U)$ is a weak solution of elliptic PDE $Lu = f$ in $U$.
Now, my main problem is in the part by setting $v = \xi^{2}u$ and performing elementary calculations, I should get $$\int_{U}\xi^{2}|Du|^{2}dx \leq C\int_{U}f^{2}+u^{2}dx$$ for some constant $C$.
How can I get the inequality above by "performing elementary calculations"? Where do I start and is there a hint?
Thank you very much and any help is much appreciated!