A Free Boundary Approach to Two-Dimensional Steady Capillary Gravity Water Waves

We prove that suitable weak solutions of the two-dimensional steady capillary gravity water wave problem (even in the presence of vorticity) are smooth. On a technical level, solutions are closely related to critical points of the Mumford–Shah functional, so that our main task is to exclude cusps pointing into the water phase. We prove that cusps do not exist, and that the free surface is smooth.