Non-existence of cusps for a free-boundary problem for water waves

Abstract

In Varvaruca and Weiss (2011), Varvaruca and Weiss eliminate the existence of cusps for a free-boundary problem for two-dimensional water waves under assumptions that hold for solutions for which $\{u>0\}$ is a “strip-like” domain in the sense of Varvaruca (2008). In this paper, it is proven that cusps do not exist in the natural setting for these free-boundary problems. In particular, non-strip-like domains are also allowed. This qualitative result follows from quantitative results which, roughly speaking, give lower bounds on the “slope” at which the free boundary approaches a stagnation point. This builds upon recent work on non-existence of cusps in McCurdy and Naples (2022) for local minimizers.