2
$\begingroup$

I have started studying Sobolev spaces and I came across a space referred to as $H^1_{loc}$. I am not sure what the $loc$ subscript infers? What is it that makes this space different from $H^1$? Why would you every specify this space instead of $H^1$?

$\endgroup$
  • $\begingroup$ Usually loc means local. So I guess $H^1_{loc}$ would mean the set of functions that are locally $L^2$ and have weak derivatives that are locally $L^2$ too. Being locally $L^2$ means that $\int_K |f|^2 < \infty$ for compact sets $K$. $\endgroup$ – desos Mar 18 '16 at 9:42
  • $\begingroup$ Where did you find such notation? Can you give us the source material? $\endgroup$ – wroobell Mar 18 '16 at 10:34
  • $\begingroup$ @jooi What is the significance of the functions being integrable on compact sets? $\endgroup$ – Riggs Mar 18 '16 at 10:42
  • $\begingroup$ @jooi: If you're working in a domain in $\Bbb R^n$, then it just means that around every point $x$ in the domain, there's a neighborhood on which the function is $H^1$. It's just about demanding sufficient integrability/differentiability hypotheses on your function locally without making such demands globally (so, eg, the integral of your function over the whole domain needn't be finite). $\endgroup$ – user98602 Mar 18 '16 at 14:48
  • $\begingroup$ Sorry, that ping was supposed to be @Riggs. $\endgroup$ – user98602 Mar 18 '16 at 17:03
0
$\begingroup$

For J.L. Lions and others, $H^m_{loc}(\Omega)$ spaces are made of functions of $H^m(\Theta)$ where $\Theta$ represents any open subset such that $\bar{\Theta}\subset \Omega$. This allows to define a family of seminorms, the norms in $H^m(\Theta)$, and in the end these spaces are Frechet spaces.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.