A min-max variational approach to the existence of gravity water waves

We establish the existence of gravity water waves by applying a mountain pass theorem to a singular perturbation of the Alt-Caffarelli functional associated with the two-dimensional water wave equations. Our approach is formulated entirely in physical coordinates and does not require the air phase to be connected, nor does it rely on symmetry or monotonicity in the $x$ or $y$ directions. The framework presented allows for both a variational approach to a variety of fluid equilibrium problems and for construction of min-max solutions to Bernoulli-type free boundary problems.