  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
