Inhomogeneous global minimizers to the one-phase free boundary problem

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 ${\mathbb{R}}^n\times [0,+\infty )$ with dilations of graphs of global minimizers $\underline{U}\le U_0\le \bar{U}$ with analytic free boundaries at distance 1 from the origin.