The dependence of the equilibrium configurations of a thin single‐crystalline epitaxial film, on the strengths of substrate‐overgrowth and overgrowth‐overgrowth bonds and the interfacial misfit is calculated neglecting substrate deformations. Previous analyses of the system were based on a one‐dimensional model, allowing for misfit, i.e., differing lattice parameters, in one interfacial direction only. The results of these analyses show (i) that interfacial mismatch is accommodated by a parallel sequence of misfit dislocations, (ii) that the mismatch decreases when appropriate overall lateral strains are introduced, (iii) that in stable and metastable film configurations both dislocations and overall strains are present, and (iv) that there exist limiting misfits 2/&pgr;l0and 1/l0, respectively, below which the stable and metastable states are pseudomorphic; the film is homogeneously strained to coherency, i.e., exact fit, with the substrate. The transition to the noncoherent state is critical, the change in average lattice parameter being discontinuous. The parameterl0depends on the various bonding strengths and yields, in an average case, the values 2/&pgr;l0=9% and 1/l0=14%. The present analysis allows for misfits along two interfacial directions, which may differ from each other and which could be accommodated by a cross grid of dislocations. Qualitatively, the results have many similarities in common with the one‐dimensional case. Quantitatively, most of the results are significantly different, reducing for example, the forementioned limiting misfits by approximately 25%. Furthermore, the transition to noncoherency is not critical and the metastable configurations are probably absent for an island with a curved boundary. Most of the differences are due to the Poisson phenomenon. Some speculations, which deal with refinements of the calculations as to overgrowth shape, size, thickness, and substrate interaction potential are presented.