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.

arXiv comment65 pages, 7 figures. In this new version, a mistake in the proof of Proposition 3.1 has been fixed

References in corpus (10)

• Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond
• Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals
• Analysis of Nematic Liquid Crystals with Disclination Lines
• On Minimizers of the Landau-de Gennes Energy Functional on Planar Domains
• Refined approximation for a class of Landau-de Gennes energy minimizers
• Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model
• Density of smooth maps for fractional Sobolev spaces $W^{s,p}$ into $\ell$ simply connected manifolds when $s\ge 1$
• Representing three-dimensional cross fields using 4th order tensors
• Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals
• Approximation in Sobolev spaces by piecewise affine interpolation

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