Topological singular set of vector-valued maps, II: $\Gamma$-convergence for Ginzburg-Landau type functionals

Abstract

We prove a $\Gamma$-convergence result for a class of Ginzburg-Landau type functionals with $\mathcal{N}$-well potentials, where $\mathcal{N}$ 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 $\pi_{k-1}(\mathcal{N})$) 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 arXiv:1712.10203.