Some remarks on the structure of finite Morse index solutions to the Allen-Cahn equation in $\mathbb{R}^2$

TL;DR

This paper proves the uniqueness of the blow-down limit for solutions to the Allen-Cahn equation in two dimensions with finite Morse index, and shows that such solutions have a multiplicity one property in their limit.

Contribution

It establishes the uniqueness of the blow-down limit and the multiplicity one property for finite Morse index solutions in 2, advancing understanding of their asymptotic behavior.

Findings

Blowing down limit is unique for solutions with linear energy growth.

Finite Morse index solutions exhibit multiplicity one in their blow-down limit.

Results contribute to the classification of solutions in 2.

Abstract

For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.