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in Why no trace operator in $L^2$? it is mentioned, that there exists a linear continuous trace operator from $L^2(\Omega)$ to $H^\frac12(\partial\Omega)$* for sufficiently smooth boundary. Can you give me any reference for this statement? I need something like this and can not find it anywhere else.

Michael
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  • I don't remember I mentioned that..... –  Apr 10 '14 at 08:31
  • But as you can see in the Link, you actually did :D – Michael Apr 10 '14 at 08:33
  • ＠Micheal: May be you can ask @Tomás directly. –  Apr 10 '14 at 08:46
  • Oh thanks, i did not see that u are not Tomás... How can i send a private message to him? Can't find any button or something like this – Michael Apr 10 '14 at 08:54
  • I guess Tomas will get a notice once someone write "at""name" in a comment (just like what I did now). So properly Tomas will receive a notice since I wrote that in the previous comment. –  Apr 10 '14 at 09:11
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    Hi @Michael. You can find the answer for your question in this book: http://www.springer.com/mathematics/analysis/book/978-3-642-65163-2 – Tomás May 12 '14 at 11:33

3 Answers3

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In Girault&Raviart's book, Thm 1.5 does not apply in this case. Indeed, the continuity of the trace operator from $W^{s,p}(\Omega)$ to $W^{s-\frac{1}{p},p}(\Omega)$ holds only if $s-\frac{1}{p}>0$ (this is implied by the assumptions).

bastienchaudet
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@Michael: I think that this result is not true, even when the boundary is smooth. I've been looking for it (for a while now...) in the literature, but unfortunately I haven't been able to find anything..! It seems that the continuity of the trace operator $\gamma:H^s(\Omega)\rightarrow H^{s-\frac{1}{2}}(\partial\Omega)$ holds iff $s>1/2$ (for smooth enough domains). Even the limit case $H^{\frac{1}{2}}(\Omega)\rightarrow L^2(\partial\Omega)$ does not work (see Lions & Magenes, same reference as Tomás', theorem 9.5).

Actually, in order to have a trace exactly in $L^p(\partial\Omega)$, $p>1$, the function has to be in some Besov space, see Cornelia Schneider's article Trace operators in Besov and Triebel-Lizorkin spaces (Corollary 3.17).

@Tomás: In the reference you suggest (Lions & Magenes), I assume you are referring to Theorem 9.4. But, when considering the trace operator, i.e. $\mu=0$, the assumption $\mu<s-\frac{1}{2}$ implies that $s$ has to be strictly greater than $\frac{1}{2}$ for the result to apply.

bastienchaudet
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I recommend you have a look into Girault&Raviart's book on Finite element methods for Navier-Stokes equations. Chapter 1 provides a survey on basic concepts on Sobolev spaces including also the pretty involved definition of the trace spaces. The result you are out for is given in Thm. 1.5.

Personally I find the presentation very accessible.

Of course, as @Tomás says, you will find everything in the book by Lions&Magenes Non-Homogeneous Boundary Value Problems and Applications in a more general setting.

Jan
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