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In article
<25039834.1162267012389.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
craig <ctcowan@xxxxxxxxxxx> wrote:
> Does there exist a smooth, non-zero, compactly supported function u on R such
> that
>
> v(x) (v defined below) can be extended to a smooth function on R. ??
>
>
> Let U denote the open set where u does not equal zero and define
>
> v(x):= (u''(x))/ u(x) on U.
Hint: Suppose u(x) = 0 for x <= 0 and u(x) is nonzero for small x
> 0. For such x, use the mean value theorem twice to see
|u''(x)/u(x)| >= |u''(x)/[u''(c_x)*x^2]|. That indicates v is
blowing up like 1/x^2 near 0.
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