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Let $\Omega$ be a bounded Lipschitz domain with boundary $\partial\Omega$.

There are two ways to define a space $H^1(\partial\Omega)$:

  1. By using charts, we can define $H^1(\partial\Omega)$ to contain functions $u\colon \partial\Omega \to \mathbb{R}$ such that $u\circ g_i \in H^1(D_i)$ for all $i$ where $g_i\colon D_i \subset \mathbb{R}^{n-1} \to \mathbb{R}$ is a chart map. The norm is the obvious norm.

  2. We can define a tangential gradient on $\partial\Omega$ as: $$\nabla_S \varphi = \nabla \varphi - (\nabla \varphi \cdot \nu)\nu$$ where $\nu$ is the unit normal vector on $\partial\Omega$ and $\nabla$ is the usual gradient. Here $\varphi$ is smooth. We can get a weak version of the tangential gradient by using the integration by parts formula on surface, let's call the weak tangential gradient $\nabla_T.$ Then we can define $H^1(\partial\Omega)$ as functions $u:\partial\Omega \to \mathbb{R}$ such that $u \in L^2(\partial\Omega)$ and $\nabla_T u \in L^2(\partial\Omega)$, and give it the obvious norm.

My question: are these definitions equivalent in some way? Do we have equivalence of norms? The second definition is not very common or popular, why is that?

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  • $\begingroup$ What is the domain of $\varphi$? $\endgroup$ – Tomás Dec 29 '13 at 18:40
  • $\begingroup$ It is $\partial\Omega.$ The formula should strictly be $\nabla_S \varphi = \nabla \tilde \varphi - (\nabla \tilde \varphi \cdot \nu)\nu$ where $\tilde \varphi$ is an extension of $\varphi$ into ambient space. $\endgroup$ – soup Dec 29 '13 at 18:44
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    $\begingroup$ I just checked the book of Ben Schweizer on PDE (german) and there the second definition is used. Basically it is introduced when consdering the Laplace Beltrami operator. Have you checked litearture on that? $\endgroup$ – Quickbeam2k1 Jan 13 '14 at 23:30
  • $\begingroup$ @Quickbeam2k1 sadly that book is not here in my library (so I can't attempt to decipher the German). I will take alook at Laplace-Beltrami.. $\endgroup$ – soup Jan 15 '14 at 20:23
  • $\begingroup$ The books is also quite new. But at least the "Laplace-Beltrami" buzzword should help $\endgroup$ – Quickbeam2k1 Jan 15 '14 at 21:42
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The two definitions are the same. For a reference see page 294 in Dziuk, Gerhard and Elliott, Charles M.. (2013) Finite element methods for surface PDEs. Acta Numerica, Vol.22 http://wrap.warwick.ac.uk/53966/1/WRAP_Elliott_DziEll13a.pdf

I'm not certain one can give an exact answer as to why the first definition is more popular. My personal opinion is that the first definition is a useful tool for showing many results about surface calculus and partial differential equations posed on surfaces. However, in the above reference, the authors manage to show many important results using the second formulation.

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