Sobolev spaces on boundaries Consider the Sobolev space $W^{s,2}=H^s$ for $s=\frac{1}{2}.$
Let $\Omega \subset \mathbb{R}^n$ be an open set with boundary $\partial\Omega$. I have seen two definitions of the space $H^s(\partial\Omega)$:
1) (From Demengel & Demengel) As the set of functions $u:\partial\Omega \to \mathbb{R}$ such that
$$\lVert{u}\rVert_{L^2(\partial \Omega)}^2 + \int_{\partial \Omega}\int_{\partial \Omega}\frac{|u(x) - u(y)|^2}{|x-y|^{n}}\;\mathrm{d}f(x)\mathrm{d}f(y) < \infty$$
where $\mathrm{d}f$ denotes the superficial density (which Demengel does not define; I guess just means the surface measure) on $\partial\Omega$.
2) (From Wloka etc) We can define $H^s(M)$ on a manifold; by using charts and partitions of unity we can transfer the norm back to the norm of Euclidean space on each patch, and so on. Then we just take $M=\partial\Omega$.
My questions:
1) Is the first definition really that simple or am I missing something?
2) Why the need for the second definition when the first one is so much simpler for a Sobolev space on the boundary? (My domain and boundaries can be as smooth as necessary). I guess I can just work with the $H^s$ norm as given in the displayed equation just like on Euclidean space and everything is good. 
3) Can someone point me to another source where such Sobolev spaces are defined in the same way as in Demengel?
Thank you
 A: 1) Yes it is that simple. It is a special case of the so called Slobodeckij norm for Sobolev and more generally Besov spaces.
2) Not as much a need as it offers you a different perspective. Note that there is a third definition: One can also define $H^{1/2}(\partial\Omega)$ as the trace space of $H^1(\Omega)$. When you generalize the definitions to other values of $p$ and $s$ in $W^{s,p}$ though, they may give you different spaces.
3) It is almost guaranteed that you will find in any textbook on Sobolev spaces. For example, look up Adams, or Triebel's books, or Grisvard's book.
A: One more remark on the definition of Sobolev spaces. Given a general metric measure space $(X,d,\mu)$, where $(X,d)$ is a metric space and $\mu$ is a locally finite Borel measure on $X$, one has at least four different definitions for Sobolev spaces, which are equivalent in very general context.
I strongly recommend the following book to those people who are interested in such a general theory:
J.Heinonen, P.Koskela, N.Shanmugalingam and J.Tyson, Sobolev spaces on metric measure spaces: an apporach based on uppper gradient, Cambridge Studies in Advanced Mathematics Series (To appear).
Except the references mentioned by Timur on SObolev spaces, I would also recommend the following book. In particular, it contains a nice treatment of the Sobolev type inequality from the point view of certain capacity estimate. (also nice see the nice trunction method used there)
Mazʹya, Vladimir: Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp. 
