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