Ginzburg-Landau relaxation for harmonic maps on planar domains into a general compact vacuum manifold

TL;DR

This paper investigates the asymptotic behavior of Ginzburg-Landau minimizers with a nonlinear potential vanishing on a manifold, showing convergence to harmonic maps with singularities and establishing related $ ext{Γ}$-convergence and regularity results.

Contribution

It generalizes known results for circle targets to arbitrary compact manifolds, analyzing the convergence and singularity structure of minimizers and solutions.

Findings

Minimizers converge to singular harmonic maps with energy concentration points.

Singularities' locations minimize a renormalized energy.

Established $ ext{Γ}$-convergence and regularity estimates for solutions.

Abstract

We study the asymptotic behaviour, as a small parameter $\varepsilon$ tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold $\mathcal{N}$ and with a given $\mathcal{N}$-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular $\mathcal{N}$-valued harmonic map, which is smooth outside a finite number of points around which the energy concentrates and whose singularities' location minimises a renormalised energy, generalising known results by Bethuel, Brezis and H\'elein for the circle $\mathbb{S}^1$. We also obtain $\Gamma$-convergence results and uniform Marcinkiewicz weak $L^2$ or Lorentz $L^2$ estimates on the derivatives. We prove that solutions to the corresponding Euler-Lagrange equation converge uniformly to the constraint and converge to harmonic maps away from singularities.