# 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.

• 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