  Given an elliptic operator $L= - \mathrm{div} (A \nabla \cdot)$ subject to
mixed boundary conditions on an open subset of $\mathbb{R}^d$, we study the
relation between Gaussian pointwise estimates for the kernel of the associated
heat semigroup, H\"older continuity of $L$-harmonic functions and the growth of
the Dirichlet energy. To this end, we generalize an equivalence theorem of
Auscher and Tchamitchian to the case of mixed boundary conditions and to open
sets far beyond Lipschitz domains. Yet we prove consistency of our abstract
result by encompassing operators with real-valued coefficients and their small
complex perturbations into one of the aforementioned equivalent properties. The
resulting kernel bounds open the door for developing a harmonic analysis for
the associated semigroups on rough open sets.
