Ginzburg–Landau Relaxation for Harmonic Maps on Planar Domains into a General Compact Vacuum Manifold

Monteil, Antonin;Rodiac, Rémy;Van Schaftingen, Jean
(2021) Archive for Rational Mechanics and Analysis — Vol. 242, n° 2, p. 875-935 (2021)

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We study the asymptotic behaviour, as a small parameter ε tends to zero, of minimisers of a Ginzburg-Landau type energy with a nonlinear penalisation potential vanishing on a compact submanifold N and with a given N-valued Dirichlet boundary data. We show that minimisers converge up to a subsequence to a singular 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 Hélein for the circle S1. We also obtain Γ-convergence results and uniform Marcinkiewicz weak L2 or Lorentz L2 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.
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Monteil, A., Rodiac, R., & Van Schaftingen, J. (2021). Ginzburg–Landau Relaxation for Harmonic Maps on Planar Domains into a General Compact Vacuum Manifold. Archive for Rational Mechanics and Analysis, 242(2), 875-935. https://doi.org/10.1007/s00205-021-01695-8 (Original work published 2021)