One-phase free-boundary problems with degeneracy

TL;DR

This paper investigates the geometric structure of free boundaries in a degenerate Bernoulli problem, revealing rectifiability and Minkowski estimates for the free boundary, inspired by water wave theories.

Contribution

It introduces a novel analysis of free boundary regularity for a degenerate functional with a distance-dependent weight, extending classical results to degenerate cases.

Findings

Free boundary decomposes into rectifiable and cusp sets.

Upper Minkowski content estimates for the free boundary.

Limited understanding of the cusp set with current techniques.

Abstract

In this paper, we study local minimizers of a degenerate version of the Alt-Caffarelli functional. Specifically, we consider local minimizers of the functional $J_{Q}(u, \Omega):= \int_{\Omega} |\nabla u|^2 + Q(x)^2\chi_{\{u>0\}}dx$ where $Q(x) = \text{dist}(x, \Gamma)^{\gamma}$ for $\gamma>0$ and $\Gamma$ a $C^{1, \alpha}$ submanifold of dimension $0 \le k \le n-1$. We show that the free boundary may be decomposed into a rectifiable set, on which we prove upper Minkowski content estimates, and a degenerate cusp set about which little can be said in general with the current techniques. Work in the theory of water waves and the Stokes wave serves as our inspiration, however the main thrust of this paper is to study the geometry of the free boundary for degenerate one-phase Bernoulli free-boundary problems in the context of local minimizers.