From: ullrich@math.okstate.edu
Newsgroups: sci.math.research
Subject: Re: Sobolev spaces
Date: Thu, 03 Dec 1998 16:45:20 GMT
In article <01be1e07$5488ac00$7a075dc0@cedrat318.cedrat-grenoble.fr>,
"hary" wrote:
> Greeting !!!
>
> I would like to know the meaning of
>
> H^(-1/2) in the Sobolev spaces theory.
>
> Thank you very much
There are a lot of context-dependent details, but
I think typically, supposing for simplicity we're talking
about functions defined on the line, a function is in H(alpha)
if "it has alpha derivatives in L^2", where "has derivative"
is defined in terms of the Fourier transform. So a function
f is in H(alpha) on the line if the integral of
|f^(x)|^2 * (1 + |x|)^alpha
is finite. (Where just for fun ^ means "Fourier transform"
and also "to the power" - I'll let you figure out which
is which.)
-----------== Posted via Deja News, The Discussion Network ==----------
http://www.dejanews.com/ Search, Read, Discuss, or Start Your Own
==============================================================================
From: Paul Metier
Newsgroups: sci.math.research
Subject: Sobolev spaces; H(-1/2).
Date: Wed, 09 Dec 1998 17:38:34 +0100
Art Werschulz wrote:
>
> Hi.
>
> "hary" writes:
>
>> I would like to know the meaning of H^(-1/2) in the Sobolev spaces
>> theory.
>
> Let $\Omega$ be a region in $d$-dimensional space. If $m$ is a
> positive integer, then $H^m_0(\Omega)$ is the set of functions having
> compact support within $\Omega$, whose Sobolev $\|\cdot\|_m$ norm is
> finite.
Correction, if you permit...
$H^m_0(\Omega)$ is the CLOSURE of the set of functions having
compact support within $\Omega$, whose Sobolev $\|\cdot\|_m$ norm is
finite.
This makes a big difference in $\Omega$ (but not in R^d).
Furthermore, one can say that $H^m_0(\Omega)$ is the set of functions in $H^m(\Omega)$ which are zero on
the border of $\Omega$.
> Then $H^{-m}(\Omega)$ is the dual space of $H^m_0(\Omega)$. Its norm
> is given by
> $$\|v\|_{-m} = \sup_{w\in H^m_0(\Omega)}
> {\bigg|\int_\Omega v(x)w(x)\,dx \over \|w\|_m},$$
> (with $0/0=0$).
>
> For non-integer values of $m$, these spaces are defined by Hilbert
> space interpolation.
I think this is correct for positiv number, but (even by duality) I'm not sure for negativ numbers.
And the question of H^(-1/2) is more complicated. You probably know that, for example,
if we set Gamma as Omega's border,
H^(1/2)(Gamma) is the set of the trace of functions in H^1(Omega)
(as more generaly, W^(m-1/p,p)(Gamma) is the set of trace of functions in W^(m,p)(Omega),
talking about more general SOBOLEV spaces)
But, talking about negativ numbers...
Even about zero, the question need to be sharper.
> For further information, you can check books on the modern theory of
> elliptic partial differential equations (e.g., Ho\"rmdander or Lions)
> or finite element methods for elliptic problems (e.g., Oden and Reddy).
>
> --
> Art Werschulz (8-{)} "Metaphors be with you." -- bumper sticker
> GCS/M (GAT): d? -p+ c++ l u+(-) e--- m* s n+ h f g+ w+ t++ r- y?
> Internet: agw@cs.columbia.eduWWW
> ATTnet: Columbia U. (212) 939-7061, Fordham U. (212) 636-6325
I'm french, that's why I can recommand you a french book talking about
H^(-1/2) in a short paragraph, but which can help you with the norm on this space.
(treating with the LAPLACE-BELTRAMI operator)
LIONS Jacque-Louis 1968 Contro^le optimal de syste'mes gouverne's par des e'quations aux de'rive'es
partielles
You will find that pages 62-63,
and a reference to LIONS-MAGENES[1] for more theoretical details.
Note: to read mathematical french, no need to speak french. Words are pratically the same as in english.
I apologize if I have been to rude in my language and explanations.
Paul METIER
Laboratoire d'Analyse Numerique
Universite Pierre et Marie Curie (Jussieu, Paris 6)
e-mail: metier@ann.jussieu.fr