For a smooth test function \eta and some constant C is it possible to have an estimate like the following?


|grad \eta|^2 < C {\eta}^2 ?


Thanks.



Test function                                  .
No. Take any line through the support of $\eta$. Along such a line, you would have
$|d\eta/ds|\le C|\eta|$. But since $\eta=0$ on a part of the line, you get $\eta=0$ everywhere by Gronwall's inequality.


