Inhomogeneous global minimizers to the one-phase free boundary problem

TL;DR

This paper constructs a foliation of the half-space using dilations of global minimizers with analytic free boundaries, providing insights into the structure of solutions to the one-phase free boundary problem.

Contribution

It introduces a foliation of the half-space by dilations of global minimizers with analytic free boundaries, expanding understanding of the solution space.

Findings

Existence of a foliation of the half-space by dilated minimizers.

Construction of bounds $$ and $ar U$ for the minimizer.

Analytic regularity of the free boundaries at a fixed distance from the origin.

Abstract

Given a global 1-homogeneous minimizer $U_0$ to the Alt-Caffarelli energy functional, with $sing(F(U_0)) = \{0\}$, we provide a foliation of the half-space $\R^{n} \times [0,+\infty)$ with dilations of graphs of global minimizers $\underline U \leq U_0 \leq \bar U$ with analytic free boundaries at distance 1 from the origin.