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 $\Gamma$.
Categories: math.AP