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Is there any way of constuct a cut off function which is $1$ in $B(0,\epsilon)$ and zero outside the ball $B(0,2\epsilon)$ and it's first and second derivative is smaller than $1/|x|^a$ , with $a<1/4$?

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For a particular $\epsilon$ or for all of them? Since the integral $\int_0^1 \frac{dt}{t^a}$ converges, we have $\int_\epsilon^{2\epsilon} \frac{dt}{t^a}\to 0$ as $\epsilon\to 0$. By the Mean Value theorem, this is inconsistent with $u(2\epsilon)-u(\epsilon)=-1$. –  user53153 Dec 16 '12 at 16:58

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