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Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with smooth boundary. Let $u \in H^1(\Omega)$. I would like a reference for the fact that the trace of $u$ on $\partial \Omega$ is in $H^\frac{1}{2}(\partial \Omega)$.

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I guess $H^{\frac 12}(\partial \Omega)$ is defined using a partition of unity $\{\beta_j\}_{1\leq j\leq N}$ subordinated to the open cover $\bigcup_{j=1}^NO_j$ of $\partial(\Omega)$. We can first show that the map $\gamma\colon\mathcal D(\mathbb R^n)\to\mathbb R^{n-1}$, $u\mapsto u(x',0)$ can be extended to a linear continuous map $\gamma \colon H^1(\mathbb R^n)\to H^{\frac 12}(\mathbb R^{n-1})$, then use local coordinates. Maybe a good reference is Adam's book, Sobolev spaces but I don't remember if these questions are treated. –  Davide Giraudo Jan 27 '12 at 21:16
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2 Answers 2

up vote 2 down vote accepted

n case anyone is curious, the answer seems to be at mipa.unimes.fr/preprints/MIPA-Preprint05-2011.pdf, Proposition 4.5. This is an excellent introduction to fractional-order Sobolev spaces.

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In some books, $H^{1/2}$ is defined to be the trace of $H^{1}$ functions, for example in Girault, V.; Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms

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But it doesn't explain why it's denoted $H^{1/2}$. To see that we need to study Sobolev spaces of type $H^s(\Omega)$, $s\in\mathbb R$. –  Davide Giraudo Feb 10 '12 at 20:07
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In case anyone is curious, the answer seems to be at mipa.unimes.fr/preprints/MIPA-Preprint05-2011.pdf, Proposition 4.5. –  Stefan Smith Feb 14 '12 at 13:04
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@user20520 You can answer your own question, for example summing up what is said in the link. –  Davide Giraudo Mar 31 '12 at 14:11
    
Unless the answer is completely wrong or a spam, no answer deserves a -1. –  timur Jun 11 '12 at 11:41
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