Statics of 90° domain structures formed in a thin tetragonal film epitaxially grown on a cubic substrate is studied theoretically on the basis of a rigorous solution of the associated elasticity problem. Inhomogeneous internal stresses existing in the strained polydomain epitaxy are calculated by the method of fictitious dislocations distributed along domain boundaries and the film/substrate interface. The calculation properly takes into account the influence of the film free surface on the field of internal stresses. For both possible periodic laminar structures, the energies of elastic strain fields existing in the film and the substrate are evaluated. These results are used to compute the equilibrium geometric parameters of periodic domain structures that minimize the total internal energy of a polydomain epitaxy. The dependencies of the equilibrium parameters on the film thickness and the relative misfit strain between the substrate and film are described. By comparing the minimum internal energies associated with various domain configurations, a diagram of equilibrium domain patterns is developed. In the range of relative misfit strains lower than some critical value, two threshold film thicknesses exist, separating three stability regions of domain patterns which correspond to a monovariant film and two periodic domain structures. In the rest of the diagram, a pattern consisting of equivalent domains with the tetragonal axes parallel to the interface is stable, irrespective of the film thickness. ©1995 American Institute of Physics.