I would like to know for which choice of boundary conditions the title statement is true.
Let $\Omega$ be a bounded Lipschitz domain in $\mathbb R^n$, for which we regard the $L^2$-de-Rham complex.
We can impose on the domain of the differentials either (i) no boundary conditions (ii) tangential boundary conditions on all of $\partial\Omega$, or (iii) impose partial tangential boundary conditions on a 2reasonable" part $\Sigma_t$ of $\partial\Omega$. The last case has been investigated by e.g., in  and .
For closed smoothly bounded domains, it is known that is known that merely locally integrable differential forms $h$ with $dh = 0$ and $\star d \star = 0$ are already smooth.  As for the $L^2$-de-Rham complex for Lipschitz bounded domains, my impression is that the smoothness of the harmonic forms is expected or even taken for granted, but I have not found an explicit statement that clarifies this.
Unfortunately, I need comparable smoothness results only for application. Maybe it is even to simple for practioners of the field to write it down explicitly. Can provide me with some (available) resources which I can cite, or a combination of theorems?
 M. Mitrea: Mixed boundary-value problems for Maxwell's equation  V. Gol'dshtein, I. Mitrea, M. Mitrea: Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds  T. Iwaniec, C. Scott, B. Stroffolini: Nonlinear Hodge Theory on Manifolds with Boundary.