We consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a matter of fact, the elliptic operator and its semigroup act simultaneously in the interior and on the boundary. We show that the elliptic operator has a bounded $\mathrm{H}^\infty$-calculus in $\mathrm{L}^p$ if the coefficients satisfy a $p$-adapted ellipticity condition. A major challenge in the proof is that different parts of the spatial domain of the operator have different dimensions. Our strategy relies on extending a contractivity criterion due to Nittka and a non-linear heat flow method recently popularized by Carbonaro-Dragičević to our setting.

29 pages, 1 figure