Re: A Question about Sobolev Spaces

From: Kanishka Perera (kperera_at_cfl.rr.com)
Date: 08/23/04 

Date: Mon, 23 Aug 2004 15:30:01 +0000 (UTC)

Many thanks. -KP

"David C. Ullrich" <ullrich@math.okstate.edu> wrote in message
news:cgcv1a$khc$1@news.ks.uiuc.edu...
> On Sun, 22 Aug 2004 20:30:10 +0000 (UTC), "Kanishka Perera"
> <kperera@cfl.rr.com> wrote:
>
> >Let Omega be a smooth bounded domain in R^n, n \ge 1 and suppose that u
is
> >in the Sobolev space H^m_0(Omega) where m is an integer greater than or
> >equal to 2. Then does it follow that |u| also belongs to H^m_0(Omega)?
Could
> >someone please tell me where I can find a proof or a counter-example?
>
> no. take n=1, m=2, omega = (-1,1), let f be smooth with compact
> support such that f(x) = x near the origin. then |f|' has a jump
> discontinuity at 0, so |f|'' has a point mass at 0, which is not
> an L^2 function.
>
> it's true for m=1; probably you knew that...
>
> >Many Thanks.
> >KP
>
>
> ************************
>
> David C. Ullrich
>
> sorry about the inelegant formatting - typing
> one-handed for a few weeks...
>

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