On the uniqueness of minimisers of Ginzburg-Landau functionals

TL;DR

This paper establishes necessary and sufficient conditions for the uniqueness of minimizers of the Ginzburg-Landau functional in multi-dimensional spaces, considering boundary data and potential convexity, and explores the structure of non-unique minimizer sets.

Contribution

It provides a comprehensive characterization of minimizer uniqueness for Ginzburg-Landau functionals under convexity and boundary conditions, including the structure of minimizer sets when non-unique.

Findings

Conditions for minimizer uniqueness established

Structure of non-unique minimizer sets characterized

Results extended to harmonic maps

Abstract

We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for $\mathbb{R}^n$-valued maps under a suitable convexity assumption on the potential and for $H^{1/2} \cap L^\infty$ boundary data that is non-negative in a fixed direction $e\in \mathbb{S}^{n-1}$. Furthermore, we show that, when minimisers are not unique, the set of minimisers is generated from any of its elements using appropriate orthogonal transformations of $\mathbb{R}^n$. We also prove corresponding results for harmonic maps