2021/11/04 by McCurdy, Sean
#35R35 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2111.03150
In this paper, we continue the study local minimizers of a degenerate version of the Alt-Caffarelli functional. Specifically, we consider local minimizers of the functional JQ(u, Ω):= ∫Ω |∇ u|2 + Q(x)2χ_\u>0\dx where Q(x) = dist(x, Γ)γ for γ>0 and Γ a submanifold of dimension 0 ≤ k ≤ n-1. Previously, it was shown that on Γ, the free boundary ∂ \u>0\ may be decomposed into a rectifiable set S, which satisfies effective estimates, and a cusp set Σ. In this note, we prove that under mild assumptions, in the case n = 2 and Γ a line, the cusp set Σ does not exist. Building upon the work of Arama and Leoni \citeAramaLeoni12, our results apply to the physical case of a variational formulation of the Stokes' wave and provide a complete characterization of the singular portion of the free boundary in complete generality in this context.