# Surjectivity of the trace operator in Sobolev spaces

Suppose $U$ is an open bounded set with $C^1$ boundary. It is a well-known result in the theory of Sobolev spaces $W^{1,p}$ that there is a continuous linear operator $T:W^{1,p}(U)\rightarrow L^p(\partial U )$ that equals ordinary restriction on continuous functions. Wikipedia tells me that this operator is in general not surjective. I know that one way to prove this is to show that functions in the image are actually more regular than your general $L^p$-function, and I have somewhat understood the proof for that. However, I feel that there should be some elementary counterexample. In fact, any function that is a trace of some $u\in W^{1,p}(U)$ is a limit of a Cauchy sequence of functions in $C^{\infty}(\bar{U})$ (and vice versa), and that could be exploited to exhibit some example where $T$ is not surjective. Does anyone have a good example?

## 2 Answers

At least in the best circumstances, it is easy to prove that the trace/restriction map loses ${\ell\over p}+\epsilon$ in "Sobolev units" (for arbitrarily small $\epsilon>0$) where codimension is $\ell$. And an easy extension of Sobolev imbedding theorems shows that (for example, for $L^2$ Sobolev spaces so I don't mess up the indexing shift...) $H^{k+{n\over 2}+\epsilon}\subset C^{k,\epsilon}$, the latter being $C^k$ functions with an $\epsilon$ Lipschitz condition.

Thus, with $p=2$ for example, $H^1(\Omega)$ maps to $C^{0,{1\over 2}-\epsilon}(\partial \Omega)$ for every $0<\epsilon<{1\over 2}$.

• Hi. Do you know any reference I can check for the case of codimension greater than 1? – yess Apr 21 '14 at 5:07
• Iterating the codimension-one case should cover the general-codimension case. – paul garrett Apr 21 '14 at 12:24

The image of $T(W^{1,p}(\Omega))$ is $L^{1}(\partial\Omega)$ if $p=1$ and the fractional Sobolev space $W^{1-1/p}(\partial\Omega)$ if $1<p<\infty$. You can find the theorems in Adams and Fournier "Sobolev spaces", Leoni "A First Course in Sobolev Spaces", Necas "Direct Methods in the Theory of Elliptic Equations". The case $p=1$ is in the last two books.

• Are you able to specify which results, exactly, in the last two cases detail the $p=1$ case, please? Do you know if any of these texts provide a direct proof - or counterexample - that when $p\ge2$ the trace is not surjective on $L^{1}(\partial\Omega)$ (I'm not so concerned atm with the fractional case)? – Sarah Paulson's Scream Jan 26 at 12:53
• I don't understand your question. When $p\ge 2$ The trace of a function in $W^{1,p}(\Omega)$ belongs to the fractional Sobolev space $W^{1-1/p,p}(\partial \Omega)$. There are many functions in $L^1(\Omega)$ which do not belong to this space. – Gio67 Jan 27 at 0:37
• Sorry, let me phrase them another way; (i) can you indicate which results from Leoni "A First Course in Sobolev Spaces" and Necas "Direct Methods in the Theory of Elliptic Equations" detail equality for the $p=1$ case, please? (ii) Is there a reference (perhaps one of the books you noted) for a standard proof or counterexample that this trace is not surjective which doesn't rely on invoking the fractional Sobolev Spaces? I understand these might sound moronic, but this is not an area with which I am very well versed - perhaps apparent. – Sarah Paulson's Scream Feb 1 at 16:39
• Secon edition in Leoni: Theorem 18.18. First edition: Theorem 15.6 for a half space. Necas 2012 edition there is only a remark for $p=1$. Remark 5.5. No proof. As for (ii) perhaps the best way is to do $p=2$ and use Fourier transforms. See Theorem 5.1 in Necas page 87. It’s giving conditions for the trace space in terms of the Fourier transform. – Gio67 Feb 2 at 11:16
• I'll check this out - thanks for getting back to me on it! – Sarah Paulson's Scream Feb 2 at 13:11