Effect of strain inhomogeneity on a chiral p-wave superconductor

Motivated by recent measurements of strain effects on the transition temperature (T$_c$) of Sr$_2$RuO$_4$, we study the strain response of a two-dimensional chiral p-wave supercoductor. We focus on the effects of inhomgeneous strain fields, which are always present in any such experiment, and which have been neglected in previous theoretical treatments. We show that the response of T$_c$ of a chiral superconductor to strain chages from being linear without inhomogeneity, to quadratic in the presence of inhomogeneity. We discuss our results in the context of the ongoing debate of superconductivity in Sr$_2$RuO$_4$.