This problem comes from a statement in the book $\textit{Elliptic Partial Differential Equations: Second Edition}$ written by Han and Lin. In its chapter 1, there's Lemma 1.41 (in the page 22) as follows.
Suppose $\{a_{ij}\}$ is a constant positive definite matrix such that for all $\xi\in\mathbb R^n,$ $$\lambda|\xi|^2\le a_{ij}\xi_i\xi_j\le \Lambda|\xi|^2$$ for some constant $0<\lambda\le\Lambda.$ Suppose $u\in C^1(B_1)$ satisfies $$\int_{B_1}a_{ij}D_iuD_j\varphi=0$$ for all $\varphi\in C^1_0(B_1),$ then there exists $c>0$ such that for all $\rho\in(0,1],$ $$\int_{B_\rho}|u|^2\le c\rho^n\int_{B_1}|u|^2.$$
In the statement above, $B_\rho\subseteq\mathbb R^n$ is the open ball centered at the origin with radius $\rho.$
My question is the first step in its proof. It says that we only need to treat the case $\rho\in(0,\frac 12]$ since the inequality is trivial for $\rho\in(\frac 12,1].$ I could not understand why it is trivial in the latter case.
Any comment is appreciated!
Perhaps the 1/2 is a red herring; it could be any constant bigger than 0. The point is that trivially, merely because $B_\rho\subset B_1$, $$ \int_{B_\rho} |u|^2 \le \int_{B_1} |u|^2. $$ The important part of the lemma is the decay rate as $\rho\to 0$: in this regime, $c\rho^n$ is an improvement over the above constant 1.