Consider the fallowing result from Evans, 2010, page 279:
Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose $\partial U$ is $C^1$. Assume $1 \leq p < n$, and $u \in W^{1,p}(U)$. Then $u \in L^{p^\ast}(U)$, with the estimate $$ ||u||_{L^{p^\ast}(U)} \leq C ||u||_{W^{1,p}(U)}, $$ the constant $C$ depending only on $p,n$ and $U$.
It's clear we can't apply the obove result for a cube, since its boundery is not $C^1$. On the other hand, once we can apply the result for a ball in $\mathbb{R}^n$ and we can put a cube inside a ball and vice versa, is it true that the result is also valid for a general cube? That is, I want to know if there's a constant $C > 0$ such that $$ ||u||_{L^{p^\ast}(Q)} \leq C ||u||_{W^{1,p}(Q)}, \forall u \in W^{1,p}(Q), $$ where $Q = B_{r}(0)$ is a cube in the maximum norm.
Yes. The proof only required $\partial \Omega$ to be $C^1$ in order to guarantee $u$ has a Sobolev extension. Though $C^1$ is sufficient for this to be true it is not necessary. It is also true if $\partial \Omega$ is Lipschitz as in the case of a cube (for references see the second paragraph on p.2206 of this paper. In fact, this proof still works on any Sobolev extension domain.