Consider the periodic strip $\Omega=\mathbb{T}\times[0,1]$ where $\mathbb{T}$ is the 1D torus with period 1. We consider the mixed Dirichlet/Neumann problem $$-\Delta u=f$$ with boundary conditions $$u(x,0)=0,\,\partial_y u(x,1)=0$$ for all $x\in \mathbb{T}$. I'd like to say that the Laplacian associated with these boundary conditions is invertible, with compact inverse with elliptic regularity estimates of the form $$\|u\|_{H^2}\le C\|f\|_{L^2}.$$ Here's what I know: i) such elliptic estimates are classical for Dirichlet and Neumann boundary conditions, separately. ii) in the case of arbitrary smooth domain $\Omega$ with homogeneous Dirichlet boundary conditions on a boundary portion $\Gamma\subset\partial\Omega$ and homogeneous Neumann boundary conditions on the remainder $\partial\Omega\setminus\Gamma$, the solution $u$ maybe be singular (say, near the boundary where there's a transition from Dirichlet to Neumann boundary conditions) so that, in particular, $u\notin H^2$.

However, for the problem that I've stated, we need not worry about a nonsmooth transition from Dirichlet to Neumann boundary conditions (the corresponding boundary portions are strictly disjoint), so I suspect that elliptic regularity maybe holds. Are there any references for such a problem?