preprint

Inserted: 11 may 2026

Year: 2026

Abstract:

We show that every entire solution to the Bernoulli (or one-phase) free boundary problem with finite Morse index in $\mathbb{R}^3$ is axially symmetric. In fact, we additionally prove that the same result would follow in any dimension $4 \le n \le 6$ in which stable entire solutions are shown to be flat.