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I am learning about Sobolev spaces on hypersurfaces.

Let $S$ be a $C^k$-hypersurface with boundary for some $k$.

In order to define a weak derivative, one needs $k \geq 2$ because the integration by parts formula that the weak derivative will satisfy includes the mean curvature term, for which we need $S \in C^2.$

We can then define $L^2(S)$ as the completion of $C^0(S)$ under the $L^2$ norm. Likewise, define $H^1(S)$ as the completion of $C^1(S)$ under the $H^1$ norm.

With this definition, it follows that $H^1(S) \subset L^2(S)$ continuously and densely.

Does all of this seem correct? I am worried about the fact that I have not said anything about the boundary of $S$ -- does it work out to define Sobolev/Lebesgue spaces in this way? I guess I may need some assumptions on the boundary but not sure what. I would appreciate any help/further details. I know this question is basic but I didn't get any reply on stackexchange...

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On a $C^1$ hypersurface, you can define the class of continuously differentiable functions in $L^2$ whose differential is also in $L^2$, and its completion in the obvious norm. That gives a definition of Sobolev space $H^1$. The theory of currents defines a notion of weak differential on any locally integrable function without specifying a norm, just on an abstract manifold without a Riemannian metric. You could then just ask, for a given Riemannian metric, whether that differential is representable by a square integrable 1-form, $H^1 \subset L^2$. I am not seeing any mean curvature term. –  Ben McKay May 18 '13 at 10:52
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