Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

Montero, Jose Alberto; Sternberg, Peter; Ziemer, William P.

doi:10.1002/cpa.10113

Cited by 28 publications

(36 citation statements)

References 28 publications

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“…In this case, our arguments will show that the associated current T * is a local minimizer of mass. This has been proved elsewhere when l = 0, under the assumption that h − is concave and h + convex near x = 0, [14,23]. Below we adopt the convention that when l = 0, R = {0}.…”

Section: Construction Of P W V Invoking the Non-degeneracy Assumptimentioning

confidence: 81%

“…It remains to verify (4.8). This follows from inspection of the argument on pages 110-111 of [23]. We give a slightly different argument here, which can and will be repeated with very few changes for every example we consider in this section.…”

Section: Some Applicationsmentioning

confidence: 99%

“…These automatically yield some results describing asymptotic behavior of minimizing sequences. Existence of local minimizers for these functionals is proved using Γ-convergence arguments in [14,23,12].…”

Section: Introductionmentioning

confidence: 99%

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Critical points via Γ-convergence: general theory and applications

Jerrard¹,

Sternberg²

2009

J. Eur. Math. Soc.

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Abstract. It is well-known that Γ-convergence of functionals provides a tool for studying global and local minimizers. Here we present a general result establishing the existence of critical points of a Γ-converging sequence of functionals provided the associated Γ-limit possesses a nondegenerate critical point, subject to certain mild additional hypotheses. We then go on to prove a theorem that describes suitable non-degenerate critical points for functionals, involving the arclength of a limiting singular set, that arise as Γ-limits in a number of problems. Finally, we apply the general theory to prove some new results, and give new proofs of some known results, establishing the existence of critical points of the 2d Modica-Mortola (Allen-Cahn) energy and 3d Ginzburg-Landau energy with and without magnetic field, and various generalizations, all in a unified framework.

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Section: Construction Of P W V Invoking the Non-degeneracy Assumptimentioning

confidence: 81%

Section: Some Applicationsmentioning

confidence: 99%

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Critical points via Γ-convergence: general theory and applications

Jerrard¹,

Sternberg²

2009

J. Eur. Math. Soc.

Self Cite

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“…e.g. [15,29]), but here the argument is in some ways more subtle due to the nonlocal term in the energy which prefers multi-component competitors. Next, we show that competitors that are uniformly close are in fact C 2 -close.…”

Section: Introductionmentioning

confidence: 99%

On the global minimizers of a nonlocal isoperimetric problem in two dimensions

Sternberg¹,

Topaloglu²

2011

Interfaces Free Bound.

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We analyze the minimization of a nonlocal isoperimetric problem (NLIP) posed on the flat 2-torus. After establishing regularity of the free boundary of minimizers, we show that when the parameter controlling the influence of the nonlocality is small, there is an interval of values for the mass constraint such that the global minimizer is exactly lamellar, that is, the free boundary consists of two parallel lines. In other words, in this parameter regime, the global minimizer of the 2d (NLIP) coincides with the global minimizer of the local periodic isoperimetric problem.

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“…Unlike monopoles, magnetic vortices not only arise as theoretical constructs but also play important roles in areas such as superconductivity (Abrikosov 1957;Ginzburg & Landau 1965;Jaffe & Taubes 1980), electroweak theory (Ambjorn & Olesen 1988, 1989a,b, 1990) and cosmology (Vilenkin & Shellard 1994). The mathematical existence and properties of such vortices have been well studied (Jaffe & Taubes 1980;Berger & Chen 1989;Neu 1990;Du et al 1992;Spruck & Yang 1992a,b;Bethuel et al 1994;Weinan 1994;Bethuel & Rivière 1995;Lin 1995Lin , 1998Ovchinnikov & Sigal 1997;Bauman et al 1998;Serfaty 1999;Pacard & Rivière 2000;Yang 2001;Montero et al 2004;Tarantello 2008;B. J. Plohr 1980, unpublished data).…”

Section: Introductionmentioning

confidence: 99%

Electrically and magnetically charged vortices in the Chern–Simons–Higgs theory

et al. 2009

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In this paper, we prove the existence of finite-energy electrically and magnetically charged vortex solutions in the full Chern-Simons-Higgs theory, for which both the Maxwell term and the Chern-Simons term are present in the Lagrangian density. We consider both Abelian and non-Abelian cases. The solutions are smooth and satisfy natural boundary conditions. Existence is established via a constrained minimization procedure applied on indefinite action functionals. This work settles a long-standing open problem concerning the existence of dually charged vortices in the classical gauge field Higgs model minimally extended to contain a Chern-Simons term.

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Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

Cited by 28 publications

References 28 publications

Critical points via Γ-convergence: general theory and applications

Critical points via Γ-convergence: general theory and applications

On the global minimizers of a nonlocal isoperimetric problem in two dimensions

Electrically and magnetically charged vortices in the Chern–Simons–Higgs theory

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