Why does the following inequality holds for any weak solution $u\in
C^1(B_1)$ of uniformly elliptic equation $D_i(a_{ij}D_ju)=0$?
Now I'm studying by Q.Han and F. Lin. Throughout the section 5 of the
chapter 1, $u\in C^1(B_1)$ is a weak solution of $$D_i(a_{ij}D_j u)=0$$
where $0<\lambda|\xi|^2\leq a_{ij}\xi_i\xi_j\leq \Lambda|\xi|^2$. In this
setting, it says that for any $0<\rho<r\leq 1$, $$\int_{B_\rho}u^2\leq
c\left(\frac{\rho}{r}\right)^\mu\int_{B_r}u^2$$ where $\mu$ depends only
on $n$, $\lambda$ and $\Lambda$. It is presented as a remark of the
following lemma: if $u$ is such a weak solution, then we have
$$\int_{B_{R/2}}u^2\leq \theta\int_{B_R}u^2$$ where
$\theta=\theta(n,\lambda,\Lambda)\in(0,1)$ and $0<R\leq1$. This lemma is
straightforward by using the Poincare inequality and the Cacciopolli
inequality. However, I have no idea how to get the first inequality from
this lemma. (The author says it follows by iterating the result of this
lemma.) Is there any one can help?
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