s^\mathrm{2d}(\xi)=
e_\mathrm{LHY}^\mathrm{2d}(\xi)/e_\mathrm{LHY}^\mathrm{3d}(\xi)$, where the 3D LHY energy is $e_\mathrm{LHY}^\mathrm{3d}(\xi) = 16 \sqrt{2} \pi \xi^{5/2}/15$. We also plot there $s^\mathrm{2d}(\xi)$ but with $e_\mathrm{LHY}^\mathrm{2d}(\xi)$ taken from Eq. (\[e2d\]) (thin meshed curve). The approximate expression for LHY term at 3D-2D crossover almost perfectly reproduces the numerical result for $\xi < 0.3$. For larger values of $\xi$, the exact formula is in the perfect agreement with the 3D expression. The agreement between quasi-2D and 3D results for values of $\xi$ such small as $\xi=0.3$ is quite surprising because the 3D formula formally applies in the limit $\xi \gg 1$.

*Droplets at 3D-2D crossover.*—Neglecting the surface energy, which is well justified for large droplets where a bulk contribution dominates, the energy of the homogeneous droplet of volume $V$ is equal to the sum of the mean-field term, $e_\mathrm{MF}$, and LHY correction, $e_\mathrm{LHY}^\mathrm{2d}$: $$\label{Ehom}
  E_\mathrm{hom} =  \frac{\varepsilon_0}{L^3} \left( e_\mathrm{MF}(\xi) + e_\mathrm{LHY}^\mathrm{2d}(\xi) \right) V,$$ where $e_\mathrm{MF}(\xi) = \beta \xi^2$, and $\beta = \varepsilon_0 L^3 \delta g/\sqrt{g_{11}g_{22}} (\sqrt{g_{11}} +\sqrt{g_{22}})^2$ .

The droplet is stable in an empty space if its pressure vanishes, $ p= -({\rm d}/{\rm d}V) E_\mathrm{hom} =0 $. Note, that $\xi$ is proportional to the density, i.e., ${\rm d}\xi/{\rm d}V = - \xi/V$. The condition for 