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.

65 pages, 7 figures. In this new version, a mistake in the proof of Proposition 3.1 has been fixed

References in corpus (2)

• Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals
• Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals

Cited by in corpus (3)

• Asymptotic behavior of minimizing $p$-harmonic maps when $p\nearrow 2$ in dimension 2
• Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold
• Limiting behavior of minimizing $p$-harmonic maps in 3d as $p$ goes to $2$ with finite fundamental group