Non-existence of cusps for degenerate Alt-Caffarelli functionals

Abstract

We eliminate the existence of cusps in a class of \textit{degenerate} free-boundary problems for the Alt-Caffarelli functional $J_Q(v,\Omega ):={\int }_{\Omega }|\nabla v{|}^2+Q^2(x){\chi }_{\{v>0\}}dx,$ so-called because $Q(x)=\text{dist}(x,\Gamma {)}^{\gamma }$ for $\Gamma$ an affine $k$-plane and $0<\gamma$. This problem is inspired by a generalization of the variational formulation of the Stokes Wave by Arama and Leoni. The elimination of cusps implies that the results of [Mccurdy20] in fact describe the entire free-boundary as it intersects $\Gamma$.