  Let $L= - \mathrm{div} (A \nabla \cdot)$ be an elliptic operator defined on
an open subset of $\mathbb{R}^d$, complemented with mixed boundary conditions.
Under suitable assumptions on the operator and the geometry, we derive an
atomic characterization (depending only on the boundary conditions) for the
Hardy space $H^1_L$ defined using an adapted square function for $L$. This
generalizes known results of Auscher and Russ in the case of pure
Dirichlet/Neumann boundary conditions on Lipschitz domains. In particular, we
develop a connection between the harmonic analysis of $L$ and its underlying
geometry.
