I am looking for a reference to point at for the following fact.
Let $\Omega_1 \subset \mathbb R^k$, $\Omega_2 \subset \mathbb R^n$ be open sets ($k, n \in \mathbb N$). Then, $$W^{1,p}(\Omega_1\times \Omega_2) \simeq W^{1,p}(\Omega_1;L^p(\Omega_2)) \cap L^p(\Omega_1;W^{1,p}(\Omega_2)),$$ as Banach spaces.
This should be essentially a matter of definitions and, moreover, it is more or less proved (in a more general context) in classical works of Strichartz [1,2] but I would really like to have a direct explicit result to point at, possibly in some more modern text on Sobolev spaces (or, more generally, on function spaces).
[1] R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech 16 (1967) 1031-1060.
[2] R. Strichartz, Fubini-type theorems, Ann. Scuola Norm.-Sci(3) 22 (1968) 399-408.