Ginzburg–Landau functionals and renormalized energy: A revised Γ -convergence approach

Motivated by the works of F. Bethuel, H. Brezis, F. Hélein [5] and of F. Bethuel, T. Rivière [6], an asymptotic analysis is carried out for minimizers of the Ginzburg-Landau functional depending on a parameter ε, in the more general case of complex line bundles with prescribed Chern class over compact Riemann surfaces. Such a functional describes a 2-dimensional abelian Higgs model and is also related to phenomena in superconductivity. A suitable renormalized energy is defined which characterizes the singularities (degree one vortices) of the limiting configuration.

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Γ-convergence of a shearlet-based Ginzburg–Landau energy

Applied and Computational Harmonic Analysis ◽

10.1016/j.acha.2020.06.004 ◽

2020 ◽

Vol 49 (3) ◽

pp. 727-770

Author(s):

Philipp Christian Petersen ◽

Endre Süli

Keyword(s):

Ginzburg Landau ◽

Γ Convergence

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Relaxation of Ginzburg–Landau functional perturbed by continuous nonlinear lower-order term in one dimension

Analysis and Applications ◽

10.1142/s0219530514500146 ◽

2014 ◽

Vol 13 (01) ◽

pp. 101-123 ◽

Cited By ~ 2

Author(s):

Andrija Raguž

Keyword(s):

Asymptotic Behavior ◽

Lower Order ◽

Order Term ◽

Lower Order Term ◽

One Dimension ◽

Young Measures ◽

Ginzburg Landau ◽

Carathéodory Function ◽

Γ Convergence

We study the asymptotic behavior as ε → 0 of the Ginzburg–Landau functional [Formula: see text], where A(s, v, v′) is the nonlinear lower-order term generated by certain Carathéodory function a : (0, 1)2 × R2 → R. We obtain Γ-convergence for the rescaled functionals [Formula: see text] as ε → 0 by using the notion of Young measures on micropatterns, which was introduced in 2001 by Alberti and Müller. We prove that for ε ≈ 0 the minimal value of [Formula: see text] is close to [Formula: see text], where A∞(s) : = ½A(s, 0, -1) + ½A(s, 0, 1) and where E0 depends only on W. Further, we use this example to establish some general conclusions related to the approach of Alberti and Müller.

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Renormalized energy for Ginzburg-Landau vortices on closed surfaces

Mathematische Zeitschrift ◽

10.1007/pl00004303 ◽

1997 ◽

Vol 225 (1) ◽

pp. 1-34 ◽

Cited By ~ 9

Author(s):

Jie Qing

Keyword(s):

Ginzburg Landau ◽

Closed Surfaces ◽

Renormalized Energy

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GINZBURG–LANDAU VORTEX LINES AND THE ELASTICA FUNCTIONAL

Communications in Contemporary Mathematics ◽

10.1142/s0219199709003284 ◽

2009 ◽

Vol 11 (01) ◽

pp. 71-107

Author(s):

ROGER MOSER

Keyword(s):

Asymptotic Behavior ◽

Second Order ◽

Ginzburg Landau ◽

Vortex Lines ◽

Γ Convergence ◽

Generalized Curves

We examine the asymptotic behavior of a family of second-order functionals arising in the theory of Ginzburg–Landau vortices. The results point towards Γ-convergence with the elastica functional for generalized curves as the limit.

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LORENTZ SPACE ESTIMATES FOR THE COULOMBIAN RENORMALIZED ENERGY

Communications in Contemporary Mathematics ◽

10.1142/s0219199712500277 ◽

2012 ◽

Vol 14 (04) ◽

pp. 1250027 ◽

Cited By ~ 3

Author(s):

SYLVIA SERFATY ◽

IAN TICE

Keyword(s):

Lorentz Space ◽

Vortex Lattice ◽

Construction Method ◽

Ginzburg Landau ◽

Landau Model ◽

One Dimensional ◽

Optimal Estimates ◽

Renormalized Energy ◽

Funct Anal ◽

Math Phys

In this paper we obtain optimal estimates for the "currents" associated to point masses in the plane, in terms of the Coulombian renormalized energy of Sandier–Serfaty [From the Ginzburg–Landau model to vortex lattice problems, to appear in Comm. Math. Phys. (2012); One-dimensional log gases and the renormalized energy, in preparation]. To derive the estimates, we use a technique that we introduced in [Lorentz space estimates for the Ginzburg–Landau energy, J. Funct. Anal. 254(3) (2008) 773–825], which couples the "ball construction method" to estimates in the Lorentz space L2,∞.

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