I am reading the book Elliptic Partial Differential Equation by Fanghua Lin and I got stuck at the lemma 1.36 ( the Caccioppoli inequality). The conditions of this lemma are: $u\in C^1(B_1)$ ($B_1$ is the unit ball in $R^n$) satisfies $$\int_{B_1}\sum{a_{ij}D_iuD_j\varphi}=0 \forall \varphi\in C_{0}^1(B_1)$$
$a_{ij}\in C^1(B_1)$ satisfies $$\lambda|\xi|^2\le\sum a_{ij}\xi_i\xi_j\le\Lambda|\xi|^2 \forall x\in B_1,\forall \xi\in R^n$$ where $\lambda,\Lambda>0$
In the proof of this lemma, the author set $\varphi=\eta^2u$ where $\eta\in C^1_0(B_1)$ and then he claims that
$$\lambda\int_{B_1}\eta^2|Du|^2\le \Lambda\int_{B_1}\eta|u||D\eta||Du|.$$
Can anyone prove the above inequality in the case $a_{ij}$ are not symmetric?
The inequality seems to be false as written. If we take $n=1$, so that $B_1$ is the interval $(-1,1)$, and $a_{11} = 1$ so that the ellipticity condition holds with $\Lambda = \lambda = 1$, and let $u(x) = x$ and $\eta(x) = 1-x^2$, then direct calculation shows that the left side of the desired inequality is $16/15$ and the right side is $8/15$. (I am not sure whether $C^1_0(B_1)$ means "vanishes on the boundary" or "compactly supported in the open ball"; if it's the latter, then modify $\eta$ slightly with a cutoff function. This will change both sides of the inequality only slightly, so it still fails.)
If you multiply the right side of the desired inequality by 2, then I can prove it in the symmetric case when $a_{ij} = a_{ji}$. For vectors $\xi, \zeta \in \mathbb{R}^n$ and $x \in B_1$, consider the symmetric quadratic form $q_x(\xi, \zeta) = \sum_{i,j} a_{ij}(x) \xi_i \eta_j$. We have $q_x(\xi, \xi) \ge \lambda |\xi|^2$ so $q_x$ is positive definite. Thus $q_x$ satisfies the Cauchy-Schwarz inequality $$|q_x(\xi,\zeta)| \le \sqrt{q_x(\xi,\xi) q_x(\zeta,\zeta)}.$$ But since $q_x(\xi, \xi) \le \Lambda |\xi|^2$ this implies $$|q_x(\xi, \zeta)| \le \Lambda |\xi| |\zeta|. \tag{*}$$
Now if $u$ is a solution and we take $\varphi = \eta^2 u$, then we have $$\begin{align*}0 &= \int q_x(Du, D[\eta^2 u]) \\ &= \int q_x(Du, 2 \eta u D\eta + \eta^2 Du) \\ &= 2\int \eta u q_x(Du, D\eta) + \int \eta^2 q_x(Du, Du).\end{align*}$$ Thus $$\begin{align*} \lambda \int \eta^2 |Du|^2 &\le \int \eta^2 q_x(Du,Du) \\ &= -2 \int \eta u q_x(Du, D\eta) \\ &= \left| -2 \int \eta u q_x(Du, D\eta) \right| \\ &\le 2 \int \eta |u| |q_x(Du, D\eta)| \\ &\le 2 \Lambda \int \eta |u| |Du| |D\eta| \end{align*}$$ using (*) in the last line.
I am not sure how to handle the non-symmetric case.