We consider elliptic t-independent second order equations
$ - \nabla_{t,x} \cdot A(x) \nabla_{t,x} U(t, x) = 0$
with bounded, measurable coefficients posed on a cylindrical domain $\mathbb{R}_{+}\times\Omega$ with a bounded base $\Omega\subset\mathbb{R}^{d}$ and complemented with mixed Dirichlet/Neumann conditions
$U=0$ (on $\mathbb{R}_{+}\times D$)
$\nu\cdot A \nabla_{t,x} U(t, x) = 0$ (on $\mathbb{R}_{+}\times (\partial\Omega\backslash D)$)
on the lateral boundary. By means of the functional calculus for bisectorial operators we
will classify all weak solutions to these equations for which either the non-tangential maximal
function of $\nabla_{t,x} U(t, x)$ or a square function associated with $\nabla_{t,x} U$ is under $L^2$
control on the whole cylinder. Surprisingly, this can be done independently of any well-posedness issues.
Based on a joint work with Pascal Auscher.