Some symmetry results for entire solutions of an elliptic system arising in phase separation

TL;DR

This paper proves that entire solutions to a specific elliptic system modeling phase separation in Bose-Einstein condensates are one-dimensional under certain monotonicity conditions, regardless of their algebraic growth at infinity.

Contribution

It establishes one-dimensional symmetry results for monotone and half-monotone solutions of an elliptic system in two spatial dimensions, extending understanding of phase separation models.

Findings

Monotone solutions are one-dimensional in two variables.

Half-monotone solutions with one monotone component are one-dimensional.

Results hold for solutions with arbitrary algebraic growth at infinity.

Abstract

We study the one dimensional symmetry of entire solutions to an elliptic system arising in phase separation for Bose-Einstein condensates with multiple states. We prove that any monotone solution, with arbitrary algebraic growth at infinity, must be one dimensional in the case of two spatial variables. We also prove the one dimensional symmetry for half-monotone solutions, i.e., for solutions having only one monotone component.