|
|
The World Wide Wade wrote:
> 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.
u''(x)/u''(c_x) may tend to 0 so I don't see your hint as conclusive.smn
|
|