We study the kinetic and thermodynamic properties of a bilayer patterning process induced by an electrohydrodynamic instability. We construct a parametric map, depending on the dielectric contrast and ratio of two film thicknesses, that describes the conditions under which hexagonally ordered pillars or holes can form when the viscosity of the upper layer is negligible. The distinct formation of arrays of pillars and holes results from the nonlinear interactions among different modes and, hence, is governed by the kinetics. The dynamic structures of pillars or holes continue to evolve to decrease the system's free energy. During this evolution, individual pillars or holes coalesce in a coarsening process until a thermodynamically stable state is reached in the form of a localized pillar, hole, or a roll structure. The selection of the pillar or hole at the final steady state represents a thermodynamic preference that can be predicted qualitatively without solving the fully nonlinear partial differential equation.