  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)]
