# Rellich–Kondrachov theorem for traces

Let $W^{1,p}(\Omega)$ be the Sobolev space of weakly differentiable functions whose weak derivatives are $p$-integrable, where $\Omega \subset \mathbb R^n$ is a domain with Lipschitz boundary. Let furthermore $\gamma$ be the trace map.

I am looking for a theorem that allows me to conclude that $\gamma \colon W^{1,p}(\Omega) \to L^q(\partial \Omega)$ is compact whenever $q < \frac{(n-1)p}{n-p}$, thus e.g. for every $q < 4$ when $p = 2$ and $n = 3$.

(Necas, p103) has such a theorem in the necessary generality. I find the proof not very accessible, however. Another proof that I am aware of (Demengel/Demengel, p167) makes stronger assumptions on the regularity of the boundary ($C^1$ rather than $C^{0,1}$). Q1: Is such a theorem proved somewhere else in the same generality? Which source would you cite for said result?

Few books bother with the case of such embeddings as far as I can tell, most only consider embeddings of the type $W^{k,p}(\Omega) \to L^q(\Omega)$. Q2: Is that because the situation I am interested in is treated in the more general context of Besov spaces (which I am not familiar with) instead?

I have collected some more references for theorems of the Rellich-Kondrachov type here.

References:

Necas, Jindrich - Direct Methods in the Theory of Elliptic Equations.

Demengel, Françoise; Demengel, Gilbert - Functional spaces for the theory of elliptic partial differential equations.

## 1 Answer

An even more general form of the result can be found in section 7 of this paper: the result applies to Sobolev extension domains (which include Lipschitz domains) and the measure on the boundary is not necessarily the surface measure. This short paper is not self-contained: some of the heavy lifting is done in other sources such as Function spaces and potential theory by Adams and Hedberg (an expensive book, unfortunately). But it does a good job of putting the results together, and outlining two ways of obtaining the desired trace theorem: with and without Besov spaces.

Warning: if you plan to investigate this subject in any depth, you will not avoid (a) Besov spaces and (b) proofs that are not very accessible. They come with the territory.

Terminological remark: I never saw compact trace theorems being named after Rellich and Kondrachov. Their names are normally attached to compact embedding theorems between function spaces on the same domain.

• Thank you; I'll have to read up on a couple of things to understand that paper. The generalisation on the domain is certainly welcome. As for the name of the theorem: Once one identifies $\gamma(W^{1,p}(\Omega))$ with $W^{1-1/p,p}(\partial \Omega)$, the question of compactness of $\gamma$ becomes a question of the compactness of the embedding $W^{1-1/p,p}(\partial \Omega) \to L^q(\partial \Omega)$. In this sense, the theorem is of Rellich-Kondrachov type, if I am not mistaken. – anonymous Dec 19 '12 at 12:55
• That's a very nice paper. – anonymous Dec 19 '12 at 17:00