I have two questions about the informations bellow that can be found in the book elliptic partial differential equations -QING HAN and FANGHUA LIN - chapter 1-pg 19
If $a_{ij} \in C(B_{1}(0))$ satisfies $\lambda|\xi|^{2}\leq a_{ij}(x)\xi_{i}\xi_{j}\leq \zeta|\xi|^{2}$ (*) for any $x \in B_{1}(0)$ and $\xi \in \mathbb{R}^{n}$ for some positive constants $\lambda$ and $\zeta$. We consider the function $u \in C^{1}(B_{1}(0))$ satisfying
$\int_\limits{B_{1}(0)} a_{ij}D_{i}uD_{j}\phi =0$ for any $\phi \in C^{1}_{0}(B^{1}(0)$.
How can i can interpret the relation given by the integral? What is the relation between (*) and the integral ?
Thanks a lot.