Topological Singular Set of Vector-Valued Maps, II: $$\varGamma $$-convergence for Ginzburg–Landau type functionals

Abstract

AbstractWe prove a $$\varGamma $$ Γ -convergence result for a class of Ginzburg–Landau type functionals with $${\mathscr {N}}$$ N -well potentials, where $${\mathscr {N}}$$ N is a closed and $$(k-2)$$ ( k - 2 ) -connected submanifold of $${\mathbb {R}}^m$$ 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 $$\pi _{k-1}({\mathscr {N}})$$ π k - 1 ( N ) ) which solves the Plateau problem in codimension k. The analysis relies crucially on the set of topological singularities, that is, the operator $${\mathbf {S}}$$ S we introduced in the companion paper [17].