Question on trace Sobolev's theorem for domain $\Omega \times (0,T)$

Question

Let $\Omega \subset \mathbb R^3$ be an open,bounded subset with a $C^2-$boundary $\Gamma$. Fix $T>0$.

Can we claim that $W^{1,2}(\Omega \times (0,T)) \hookrightarrow C([0,T];L^2(\Gamma))(*)$

I believe it could be true since $W^{1,2}(\Omega \times (0,T)) \simeq W^{1,2}([0,T];W^{1,2}(\Omega))$ and additionally by the usual trace map theorem we have $W^{1,2}(\Omega)\hookrightarrow L^2(\Gamma)$. Finally we know that if $f\in W^{1,2}([0,T];W^{1,2}(\Omega))$ then also $f\in C([0,T];W^{1,2}(\Omega))$.

Although I have most of the pieces of the puzzle in my mind, I can't connect them in a proper mathematical way to justify my claim. What is more, if $(*)$ is indeed true, is this embedding compact?

Any help or hint, or even counterexample is much appreciated.

Thanks in advance!

Answer 1

I have realized that it is quite easy to see that this embedding fails. Let us consider the one-dimensional situation $\Omega = (0,1)$ and $T = 1$.

Then, $L^2(\partial\Omega)$ is essentially $\mathbb R^2$. Your question about regularity translates into: Does $u(\cdot,0), u(\cdot,1) \in C([0,1])$ for $u \in H^1((0,1)^2)$? This is obviously false since a function in $H^1((0,1)^2)$ can have a singularity/discontinuity at the boundary.