Quantization effects for ??u = u(1 ? |u|2) in ?2

Brezis, Haı̈m; Merle, Frank; Rivière, Tristan

doi:10.1007/bf00375695

Cited by 101 publications

(132 citation statements)

References 2 publications

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“…The first results of this character were proven by Bethuel, Brezis, and Hélein [2], using lower bounds established by Brezis, Merle, and Rivière [4]. Other important contributions include papers of Struwe [11], Bethuel and Rivière [3], Han and Li [6], and Hong [7].…”

Section: Introductionmentioning

confidence: 99%

Lower Bounds for Generalized Ginzburg--Landau Functionals

Jerrard

1999

SIAM J. Math. Anal.

167

236

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Abstract. We study properties of Ginzburg-Landau functionals I U (·), defined for functions u ∈ W 1,n (U ; R n ), where U ⊂ R n . In particular, we establish lower bounds relating the energy I U (u) to the Brouwer degree of u, and we prove under additional hypotheses that the energy concentrates on a small number of small sets. As a consequence we deduce some compactness theorems. Such estimates are useful in studying Ginzburg-Landau-type PDEs associated with the functional I U .

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Section: Introductionmentioning

confidence: 99%

Lower Bounds for Generalized Ginzburg--Landau Functionals

Jerrard

1999

SIAM J. Math. Anal.

167

236

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“…The technique thus relies on some adaptations of the Pohozaev identities with error term f ε . Observe that Pohozaev identities have already been widely used in the context of Ginzburg-Landau statics and dynamics ( [BMR,BBH,BCPS,RuS,SS2]). Some similar results and the "balanced cluster" condition (1.7) also appear in the recent preprint of Bethuel-Orlandi-Smets [BOS] (see Theorem 5) on the parabolic GinzburgLandau equation.…”

Section: Presentation Of the Problemmentioning

confidence: 99%

Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation

Serfaty

2007

J. Eur. Math. Soc.

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Abstract. We study vortices for solutions of the perturbed Ginzburg-Landau equations u + (u/ε 2 )(1 − |u| 2 ) = f ε where f ε is estimated in L 2 . We prove upper bounds for the GinzburgLandau energy in terms of f ε L 2 , and obtain lower bounds for f ε L 2 in terms of the vortices when these form "unbalanced clusters" where i d 2 i = ( i d i ) 2 . These results will serve in Part II of this paper to provide estimates on the energy-dissipation rates for solutions of the Ginzburg-Landau heat flow, which allow one to study various phenomena occurring in this flow, including vortex collisions; they allow in particular extending the dynamical law of vortices beyond collision times.

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“…It is easy to check that u ′ n solves −∆u ′ n = u ′ n (1 − |u ′ n | 2 ). Following [7], up to a subsequence, u…”

Section: Each ρ-Bad Disc Contains Exactly One Small Bad Discmentioning

confidence: 99%

“…Assume first that [7], noting that the degree of u 0 on large circles centered in 0 is 0, we obtain that u 0 = Cst ∈ S 1 and consequently…”

Section: Each ρ-Bad Disc Contains Exactly One Small Bad Discmentioning

confidence: 99%

The Ginzburg–landau Functional With a Discontinuous and Rapidly Oscillating Pinning Term. Part I: The Zero Degree Case

Santos

Mironescu

Misiats

2011

Commun. Contemp. Math.

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We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree d > 0. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and Hélein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. In some special cases we prove that their macroscopic location tends to minimize the renormalized energy of Bethuel-Brezis-Hélein. In addition, impurities affect the microscopic location of vortices. Our technics allows us to work with impurities having different size. In this situation we prove that vortices are pinned by the largest impurities.

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Quantization effects for ??u = u(1 ? |u|2) in ?2

Cited by 101 publications

References 2 publications

Lower Bounds for Generalized Ginzburg--Landau Functionals

Lower Bounds for Generalized Ginzburg--Landau Functionals

Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. Part I: Study of the perturbed Ginzburg–Landau equation

The Ginzburg–landau Functional With a Discontinuous and Rapidly Oscillating Pinning Term. Part I: The Zero Degree Case

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