Non-existence of cusps for degenerate Alt-Caffarelli functionals

We eliminate the existence of cusps in a class of \textit{degenerate} free-boundary problems for the Alt-Caffarelli functional $J_{Q}(v, Ω):= \int_Ω|\nabla v|^2 + Q^2(x)χ_{\{v>0\}}dx,$ so-called because $Q(x) = \text{dist}(x, Γ)^γ$ for $Γ$ an affine $k$-plane and $0< γ$. 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 $Γ$.

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