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



Sobolev spaces on boundaries
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.


