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In this paper we establish the incompressible limit for the compressiblefree-boundary Euler equations with surface tension in the case of a liquid.Compared to the case without surface tension treated recently, the presence ofsurface tension introduces severe new technical challenges, in that severalboundary terms that automatically vanish when surface tension is absent nowcontribute at top order. Combined with the necessity of producing estimatesuniform in the sound speed in order to pass to the limit, such difficultiesimply that neither the techniques employed for the case without surfacetension, nor estimates previously derived for a liquid with surface tension andfixed sound speed, are applicable here. In order to obtain our result, wedevise a suitable weighted energy that takes into account the coupling of thefluid motion with the boundary geometry. Estimates are closed by exploiting thefull non-linear structure of the Euler equations and invoking several geometricproperties of the boundary in order to produce some remarkable cancellations.We stress that we do not assume the fluid to be irrotational.

On the Incompressible Limit for the Compressible Free-Boundary Euler Equations with Surface Tension in the Case of a Liquid