Is the following a good definition for a Sobolev space on a boundary:
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..
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.