Finite index solutions to the Bernoulli problem in three dimensions are axially symmetric

TL;DR

This paper proves that all finite Morse index solutions to the Bernoulli free boundary problem in three dimensions are axially symmetric, extending to higher dimensions under certain stability conditions.

Contribution

It establishes axial symmetry for finite Morse index solutions in three dimensions and suggests similar results in higher dimensions with stability assumptions.

Findings

All finite Morse index solutions in $ extbf{R}^3$ are axially symmetric.

The result extends to dimensions 4 to 6 under stability assumptions.

Stable entire solutions are shown to be flat in certain dimensions.

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.