On the uniqueness of minimisers of Ginzburg-Landau functionals

We provide necessary and sufficient conditions for the uniqueness of minimisers of the Ginzburg-Landau functional for Rⁿ-valued maps under a suitable convexity assumption on the potential and for H¹⁼² \ L¹ boundary data that is non-negative in a fixed direction e 2 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 Rⁿ. We also prove corresponding results for harmonic maps with values into S^n-1