Flat-implies-Lipschitz conjecture with a narrow cylindrical hole

Set

$$\mathbb{B}_r=\{x\in\mathbb{R}^3:|x|<r\},\qquad \mathcal{C}_r=\{x\in\mathbb{R}^3:x_2^2+x_3^2<r^2\}.$$

Let $u$ be a classical solution to the one-phase free-boundary problem in $\mathbb{B}_1\setminus\mathcal{C}_\delta$, with contractible positive and zero phases. Flat-implies-Lipschitz conjecture with a cylindrical hole. There are absolute constants $\delta_0>0$, $\epsilon>0$, and $C$ such that, whenever $\delta\le\delta_0$ and, for every $x=(x_1,x_2,x_3)\in\mathbb{B}\setminus\mathcal{C}_\delta$,

$$x_1>\epsilon\implies u(x)>0,\qquad x_1<-\epsilon\implies u(x)=0,$$

the free boundary satisfies

$$F(u)\cap\mathbb{B}_{1/C}\cap\{x_2^2+x_3^2>(C\delta)^2\}=\{x\in\mathbb{B}_{1/C}:x_1=g(x_2,x_3),\ x_2^2+x_3^2>(C\delta)^2\}$$

for some $g:\mathbb{R}^2\to\mathbb{R}$ with $|\nabla g|\le 1/100$.

This extends the flat-implies-Lipschitz principle of Alt and Caffarelli to a domain with a narrow cylindrical hole and contractible phases. The source notes that the formulation is intended separately for each half of the product double-hairpin solution.