SrTiO$_3$ is a superconducting semiconductor with a pairing mechanism that is not well understood. SrTiO$_3$ undergoes a ferroelastic transition at $T=$ 105 K, leading to the formation of domains with boundaries that can couple to electronic properties. At two-dimensional SrTiO$_3$ interfaces, the orientation of these ferroelastic domains is known to couple to the electron density, leading to electron-rich regions that favor out-of-plane distortions and electron-poor regions that favor in-plane distortion. Here we show that ferroelastic domain walls support low energy excitations that are analogous to capillary waves at the interface of two fluids. We propose that these capillary waves mediate electron pairing at the LaAlO$_3$/SrTiO$_3$ interface, resulting in superconductivity around the edges of electron-rich regions. This mechanism is consistent with recent experimental results reported by Pai et al. [PRL $bf{120}$, 147001 (2018)]
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