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In Lemma 1.41 suppose $u\in C^1 (B_1)$ satisfies

$$\int_{B_1} \sum_{ij=1}^{n}a_{ij}D_i u D_j \phi =0 \ \text{for any}\ \phi\in C_0^1 (B_1) $$

Then for any $0<\rho\leq r$, there holds

$$\int_{B_{\rho}}|u|^2\leq c(\frac{\rho}{r})^n\int_{B_{r}}|u|^2$$

In the proof he said when $r=1$, the result is trivial for $\rho\in (\frac{1}{2}, 1]$. Who can explain this? Thanks a lot!

Kira Yamato
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1 Answers1

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If $\rho\in (1/2,1]$, the stated inequality holds with $c=2^n$: $$ \int_{B_{\rho}}|u|^2\leq \int_{B_{r}}|u|^2 \le c(\frac{\rho}{r})^n\int_{B_{r}}|u|^2$$ since $c(\frac{\rho}{r})^n\ge 1$.

The effect of constant factor $c$ is that the inequality becomes interesting only when $\rho$ is much smaller than $r$.