On the uniqueness of minimisers of Ginzburg-Landau functionals

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

This new version expands the previous one, in particular, we add Theorem 1.7 for the uniqueness of the minimizer when the boundary data is the equator map in dimensions $> 6$