# Asymptotic behavior of critical points of an energy involving a loop-well potential

## TL;DR

This paper investigates the asymptotic behavior of critical points of an energy functional with a loop-well potential as the parameter approaches zero, extending previous work by removing symmetry assumptions.

## Contribution

It introduces new analytical tools to analyze critical points of energy functionals with non-symmetric loop-well potentials, broadening understanding beyond symmetric Ginzburg-Landau models.

## Key findings

- Describes asymptotic behavior of critical points as epsilon approaches zero.
- Develops novel methods applicable to non-symmetric potentials.
- Extends classical results to more general loop-well potentials.

## Abstract

We describe the asymptotic behavior of critical points of $\int_{\Omega} [(1/2)|\nabla u|^2+W(u)/\varepsilon^2]$ when $\varepsilon\to 0$. Here, $W$ is a Ginzburg-Landau type potential, vanishing on a simple closed curve $\Gamma$. Unlike the case of the standard Ginzburg-Landau potential $W(u)=(1-|u|^2)^2/4$, studied by Bethuel, Brezis and H\'elein, we do not assume any symmetry on $W$ or $\Gamma$. In order to overcome the difficulties due to the lack of symmetry, we develop new tools which might be of independent interest.