# Compact embedding into boundary

I was looking for a result:

$W^{1,2}(Q)$ is compactly embedded in $L^{2}(\partial Q)$ ; where Q $\subset \mathbb R^{2}$ is a bounded domain & $Q \in C^{1,1}$." (which is mentioned in the book: "Weak & Measure Valued Solutions to Evolutionary PDEs" by Malek, Necas et al. at page-159 .

I know the result by Relich-Kondrachov Theorem that: $W^{1,2}(Q)$ is compactly embedded in $L^{2}(Q)$ & I saw here the following two links:

Now, in the above Wikipedia link, under the "An Overview" section, in the "Here are some sources", No. #3 point does not hold since $N=2$ & $k=2$ here.

So, I am confused. Can anyone help in this issue??

P.S. :- I know the trace map $H^{1}(Q) \to L^{2}(\partial Q)$ is bounded, linear & injective BUT NOT surjective. Does it have any role here??

## 1 Answer

The #3 of that Wikipedia talk page requires $1<p<N$ and $$1 \ge \frac1q > \frac1p - \frac{1}{N-1}\frac{p-1}{p}$$ So, as written, it does not apply to $p=2=N$. But on a bounded domain, $W^{1,2}$ continuously embeds into $W^{1,2-\epsilon}$, and since the inequality $$\frac12 > \frac1{2-\epsilon} - \frac{1-\epsilon}{2-\epsilon}$$ holds for small $\epsilon>0$, the result applies to the trace operator $W^{1,2-\epsilon}\to L^2$.

More directly (and with a more authoritative reference that a Wikipedia page), you can apply Corollary 7.4 of Biegert's paper which is linked from the answer you already saw.