Topological singular set of vector-valued maps. II: \( \varGamma \)-convergence for Ginzburg-Landau type functionals. (English) Zbl 1467.58004
Summary: We prove a \(\varGamma \)-convergence result for a class of Ginzburg-Landau type functionals with \({\mathscr{N}} \)-well potentials, where \({\mathscr{N}}\) is a closed and \((k-2)\)-connected submanifold of \({\mathbb{R}}^m\), in arbitrary dimension. This class includes, for instance, the Landau-de Gennes free energy for nematic liquid crystals. The energy density of minimisers, subject to Dirichlet boundary conditions, converges to a generalised surface (more precisely, a flat chain with coefficients in \(\pi_{k-1}({\mathscr{N}}))\) which solves the Plateau problem in codimension \(k\). The analysis relies crucially on the set of topological singularities, that is, the operator \({\mathbf{S}}\) we introduced in the companion paper [the authors, Calc. Var. Partial Differ. Equ. 58, No. 2, Paper No. 72, 40 p. (2019; Zbl 1411.58004)].
MSC:
| 58C06 | Set-valued and function-space-valued mappings on manifolds |
| 49Q15 | Geometric measure and integration theory, integral and normal currents in optimization |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Citations:
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