# 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

• Your first definition doesn't depend on $s$? – paul garrett Dec 20 '13 at 19:03
• Sorry for the mistake. I fix $s=\frac{1}{2}$. – soup Dec 20 '13 at 19:37
• What is superficial density? – smyrlis Dec 20 '13 at 20:13
• @smyrlis The author does not define it unfortunately.. – soup Dec 20 '13 at 20:37
• It is probably the standard $n-1$-dimensional measure of the (sufficiently smooth) surface. – smyrlis Dec 20 '13 at 20:40

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.

• Thank you. Does the norm in my displayed equation make the space Hilbert (it is not stated in Demengel, which makes me somewhat concerned)? as for point 3), I tried Adams and Grisvard but no luck. (I wonder if you missed that the domain $\partial\Omega$ is a closed hypersurface as opposed to open set..?) – soup Dec 21 '13 at 19:35
• @soup: Yes it does, because the norm is equivalent to the other standard norms. For point 3), one can localize this double integral norm, so it does not make a difference whether $\partial\Omega$ is a closed hypersurface or an open set. Have you looked at MacLean? – timur Dec 23 '13 at 19:58

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.

• I don't think that the definition of Sobolev spaces and metric spaces is not what the OP has in mind… – Dirk Dec 21 '13 at 22:43
• My comment is addressing the his second definition of Sobolev spaces. If one imposes smooth condition on M, then one can naturally define the Sobolev spaces via local coordinate. In this manner, the foudamental theorem of calculus is implicitely used. In the non-smooth case (on metric measure spaces), one can define the Sobolev spaces in a similar manner. This is certainly non-trivial and relies on the recent development on non-smooth calculus. – Changyu Guo Dec 21 '13 at 22:49
• Thanks, this is interesting but a bit above what I need. – soup Dec 27 '13 at 15:23