On the uniqueness of minimisers of Ginzburg-Landau functionals. (English) Zbl 1445.35011
Summary: 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 with values into \(\mathbb{S}^{n-1}\).
MSC:
35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
35B06 | Symmetries, invariants, etc. in context of PDEs |
35J50 | Variational methods for elliptic systems |
35Q56 | Ginzburg-Landau equations |