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Is the following a good definition for a Sobolev space on a boundary: enter image description here

Can anyone show me another source where such a space is defined?

In the definition, $v \in W^{s,p}(\partial\Omega)$ if $v \circ g_i \in W^{s,p}(D_i)$. Now, does this only need to hold for one such representation $\{g_i\}$ (as the author suggests) or all possible representations? I thought the latter..

soup
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1 Answers1

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The main question is: if we have another representation, let's say, $\{h_i\}$. What is the relation between the space $X=W^{s,p}(\partial\Omega)$ generated by $\{g_i\}$ and the space $Y=\tilde{W}^{s,p}(\partial\Omega)$ generated by $\{h_i\}$.

The answer is: $X$ is continuously embedded in $Y$ and $Y$ is continuously embedded in $X$.

For references, see chapter 6 and references therein of Function Spaces by A. Kufner, Oldrich John, Svatopluk Fucik.

Tomás
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    Just to add: That the change of variables are "ok" is embedded in the assumption that the representations $g_i$ are $C^{k,\alpha}$ and that we are looking at Sobolev spaces with $s \leq k+\alpha$. For $k+\alpha$ integral, this follows from the classical chain rule for derivatives. – Willie Wong Dec 19 '13 at 12:47
  • Thanks for the answer. Can I not just use Sobolev space $W^{s,p}(M)$ defined on manifold (of which there are many sources) and take $M=\partial\Omega$?

    And since $\partial\Omega \subset \mathbb{R}^n$, your post http://math.stackexchange.com/questions/486729/equivalence-of-norms-in-ws-p-partial-omega suggests that we can use the same definiton of $W^{s,p}(\Omega)$ except replace the integration area by $\partial\Omega$. Is that not good?

     – soup Dec 19 '13 at 13:29
  • If I understood your question, these definitions are the same ( in the sense that they are at least equivalent). Please, consider read the book I have cited and also Necas book which is cited there. – Tomás Dec 19 '13 at 13:33
  • Thanks, I will do. It is not in my library so I will have to search a bit – soup Dec 19 '13 at 13:44
  • @soup, take a look in this English version of Necas book (There is a French version, which I prefer, but I could not find it) in chapter 3, first section: http://carlossicoli.free.fr/N/Necas_J.-Direct_Methods_in_the_Theory_of_Elliptic_Equations_-Springer(2011).pdf – Tomás Dec 19 '13 at 13:48
  • Thank you @Tomás, I will read it. Demengel and Demengel, page 130 also defines the space as just an integral over the boundary using the surface measure, FYI. – soup Dec 19 '13 at 14:11