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.
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$\begingroup$ @Michael Does this mean $|{\rm grad}\; \eta(x)|^2 < C \eta^2(x)\;$ for all $x\in \mathbb R^n$? $\endgroup$– AndrewCommented Sep 7, 2011 at 20:36
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$\begingroup$ @Michael It is not valid if $\eta(x)=0$ and ${\rm grad}\eta(x)\ne0$ for some $x$. $\endgroup$– AndrewCommented Sep 7, 2011 at 21:33
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$\begingroup$ @Andrew. You are right. The condition on \eta , which I didn't include, is that it is supported in some ball B_2, identically 1 in the ball B_1 and |grad \eta| < constant. $\endgroup$– MichaelCommented Sep 13, 2011 at 0:58
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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.
answered Sep 7, 2011 at 20:36
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$\begingroup$ Thanks. Does the Gronwall's inequality hold in any ball or only in a real line( as Wikipedia) has it? Thanks again. $\endgroup$– MichaelCommented Sep 7, 2011 at 20:55