Abstract We provide the first construction of overhanging gravity water waves having the approximate form of a disk joined to a strip by a thin neck. The waves are solitary with constant vorticity, and exist when an appropriate dimensionless gravitational constant g>0 g > 0 is sufficiently small. Our construction involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an exceptional domain discovered by Hauswirth, Hélein, and Pacard (Pac. J. Math. 250: 319–334, 2011). The method developed here is related to the construction of constant mean curvature surfaces through gluing, and can be applied to other overdetermined elliptic problems.
Dávila et al. (Tue,) studied this question.