The monograph "Non-Homogeneous Boundary Value Problems and Applications" by Lions and Magenes is infamous for developing a truly extensive regularity theory for elliptic problems on domains, and for doing so under amazingly restrictive assumptions on the regularity of the boundary - basically, the domain is always assumed to be $C^\infty$.

Here comes one of their classical results, obtained by combining Theorems 2.7.3 and 2.7.4 in their first volume, when specialized to the case I am interested in:

Theorem.The trace operator $\gamma$ is bounded from $$ > D^\frac{1}{2}(\Omega):=\left\lbrace > u\in H^\frac{1}{2}(\Omega): \Delta > u\in > \Xi^{-\frac{3}{2}}(\Omega)\right\rbrace > $$ to $L^2(\partial \Omega)$, and furthermore $$\begin{pmatrix} \Delta \atop > \gamma\end{pmatrix} $$ is an isomorphism from $D^\frac{1}{2}(\Omega)$ to $\Xi^{-\frac{3}{2}}(\Omega)\times > L^2(\partial \Omega)$.

(Here $\Xi^{-\frac{3}{2}}(\Omega)$ is a rather ugly interpolation space, which however is nice enough to contain $L^2(\Omega)$).

In particular, it follows that

Corollary.$u\in H^\frac{1}{2}(\Omega)$ whenever $u$ satisfies $\Delta u=f$ for some $f\in > L^2(\Omega)$.

Many results of Lions-Magenes' have been extended to the case of $C^{1,1}$-domains, or even to general convex bounded domains, most notably in Grisvard's "Elliptic problems in nonsmooth domains", but I was not able to find an extension of the above Corollary. What I am interested in is simply the case of the hypercube $\Omega:=(0,1)^N$, that is I am asking the following

Question.Let $u$ solve $\Delta u=f$ for some $f\in L^2\left((0,1)^N\right)$. Is it true that $u\in H^\frac{1}{2}\left((0,1)^N\right)$?