I want to prove Caccioppoli inequality:
$\int_U \eta^2 |Du|^2dx\leq C \int_U |D\eta|^2u^2dx$.
I proved this result by using the definition of weak solutions by taking $v=\eta^2 u$,
I proved my result upto here
$\int_U \eta^2 |Du|^2dx\leq \frac{2M}{\theta}\int_U \eta |u| |Du||D\eta|dx$.
I tried to use Holder inequality further to reach desire inequality.
Can anyone suggest me the last step.
Use
$$2 \eta |u| |Du||D\eta| =2(\eta |Du|)(|u||D\eta)\le\epsilon \eta ^2 |Du|^2 +|u|^2|D\eta|^2/\epsilon.$$
Now choose $\epsilon$ so that $$ \frac{M\epsilon}{\theta} =\frac 12.$$
Then $$\int _U \eta^2 |Du|^2 \le \frac {M}{\theta \epsilon} \int _U u^2 |D\eta |^2 +\frac 12 \int _U \eta^2|Du|^2$$ and your inequality is obtained by moving the second term on the right to the left.