Gradients in interfacial tension at a solid-air or solid-solid interface exert tangential stresses which may deform the solid if it is sufficiently soft. We report a theoretical study of this issue by considering the case of spatially periodic interfacial tension gradients that have been patterned onto planar elastomeric layers. Assuming that the gradients are weak, we apply regular perturbation theory, Taylor series, and the theory of linearized elasticity to determine the leading order correction to the interfacial deformation. For solid-air interfaces, is found that the maximum vertical interfacial displacement occurs when the characteristic wavelength of the gradients is of the same order of magnitude as the solid thickness. For solid-solid interfaces, the maximum vertical interfacial displacement occurs when the characteristic wavelength of the gradients is on the order of the thickness of the thinner solid. The effects of varying the thickness and modulus ratios of the solids are also reported. The results suggest that patterned interfacial tension gradients may serve as a novel route for the creation of topographically patterned surfaces and interfaces in soft materials.