Conditions for Eliminating Cusps in One-phase Free-boundary Problems with Degeneracy

Abstract

In this paper, we study the geometry of the free-boundary arising from local minimizers of a degenerate version of the Alt–Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u,\Omega) := \int_{\Omega} |\nabla u|^{2} + Q(x)^{2} \chi_{\{u>0\}} \, dx$ where $Q(x) = \operatorname{dist}(x,\Gamma)^{\gamma}$ for $\gamma \gt 0$ and $\Gamma$ a submanifold of dimension $0 \leq k \leq n-1$. Previously, it was shown that on $\Gamma$, the free boundary $\partial \{u \gt 0\}$ may be decomposed into a rectifiable set $\mathcal{S}$, which satisfies effective estimates, and a cusp set $\Sigma$ [11]. In this note, we prove that under mild assumptions, in the case $n = 2$ and $\Gamma$ a line, the cusp set $\Sigma$ does not exist. Building upon the work of Arama and Leoni [3], our results apply to the physical case of a variational formulation of the Stokes' wave.