Non-existence of cusps for degenerate Alt-Caffarelli functionals

TL;DR

This paper proves that cusps do not form in a class of degenerate free-boundary problems related to the Alt-Caffarelli functional, extending understanding of the structure of free boundaries in these problems.

Contribution

It demonstrates the non-existence of cusps in degenerate free-boundary problems for a generalized Alt-Caffarelli functional, clarifying the structure of the free boundary.

Findings

Cusps are proven not to exist in the studied class of problems.

The results extend previous work to include degenerate cases with variable coefficients.

The entire free boundary's intersection with a reference plane is characterized.

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 $\Gamma$.