Estimates of the upper critical field for the Ginzburg–Landau equations of superconductivity

Lu, Kening; Pan, Xing‐Bin

doi:10.1016/s0167-2789(98)00246-2

Cited by 81 publications

(119 citation statements)

References 9 publications

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“…• We have in particular obtained a complete proof of the basic Proposition 3.1 in [LuPa1] with actually an improvement of the right hand side and an extension of the regime of parameters (κ, H) for which the estimate is true.…”

Section: Lemma 24mentioning

confidence: 82%

“…The number Θ 0 (also denoted by β 0 by some authors) plays an important role in the analysis of the Ginzburg-Landau system (see for instance [LuPa1,PiFeSt,HeMo2] for information on this spectral constant). Here we will only recall the basic property that Θ 0 ∈ (0, 1).…”

Section: Asymptotic Estimatesmentioning

confidence: 99%

“…It is known that for a given κ there exists H C3 (κ) such that for all H > H C3 (κ), the unique minimizer of E κ,H is the configuration (ψ, A) = (0, F) (up to change of gauge). This critical field H C3 (κ) has been intensively studied [GiPh,LuPa1,PiFeSt,HePa,FoHe3] and it is known to satisfy…”

Section: Proposition 42mentioning

confidence: 99%

“…In the literature such estimates are found in varying generality scattered over different publications (c.f. [LuPa1,LuPa2,LuPa3,LuPa4], [HePa],...).…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 3.1 or [LuPa1, Proposition 3.1], are the basic input to control a 'blow-up' approach as in [LuPa1,Section 4] or [HePa,Section 4]. In Section 4 we describe this approach and give the main results, see Propositions 4.2, 4.4 and 4.5.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Optimal uniform elliptic estimates for the Ginzburg-Landau system

Fournais¹,

Helffer²

2007

Adventures in Mathematical Physics

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Section: Lemma 24mentioning

confidence: 82%

Section: Asymptotic Estimatesmentioning

confidence: 99%

Section: Proposition 42mentioning

confidence: 99%

“…In the literature such estimates are found in varying generality scattered over different publications (c.f. [LuPa1,LuPa2,LuPa3,LuPa4], [HePa],...).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal uniform elliptic estimates for the Ginzburg-Landau system

Fournais¹,

Helffer²

2007

Adventures in Mathematical Physics

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On the Ginzburg‐Landau critical field in three dimensions

Fournais

Helffer

2008

Comm Pure Appl Math

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Abstract. We study the three dimensional Ginzburg-Landau model of superconductivity. Several 'natural' definitions of the (third) critical field, H C 3 , governing the transition from the superconducting state to the normal state, are considered. We analyze the relation between these fields and give conditions as to when they coincide. An interesting part of the analysis is the study of the monotonicity of the ground state energy of the Laplacian, with constant magnetic field and with Neumann (magnetic) boundary condition, in a domain Ω. It is proved that the ground state energy is a strictly increasing function of the field strength for sufficiently large fields. As a consequence of our analysis we give an affirmative answer to a conjecture by Pan.

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Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field

Alouges¹,

Bonnaillie‐Noël²

2006

Numerical Methods Partial

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International audienceMotivated by questions arising in the theory of superconductivity, we are interested in the study of the fundamental state of the Schrodinger operator with magnetic field in a domain with corners. Although this problem has been extensively studied theoretically, many fewer articles deal with numerical approaches. In this article, we propose numerical experiments based on the finite element method to determine the bottom of the spectrum of the operator. Analyzing the drawbacks of a standard method and the properties of the operator, we propose a natural gauge-invariant method and provide a few numerical simulations. We furthermore improve the numerical results by coupling the method with a mesh-refinement technique based on a posteriori error estimates developed by one of the authors. This allows us to look at the monotonicity of the smallest eigenvalue in an angular sector with respect to the angle, which complement theoretical studie

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Estimates of the upper critical field for the Ginzburg–Landau equations of superconductivity

Cited by 81 publications

References 9 publications

Optimal uniform elliptic estimates for the Ginzburg-Landau system

Optimal uniform elliptic estimates for the Ginzburg-Landau system

On the Ginzburg‐Landau critical field in three dimensions

Numerical computations of fundamental eigenstates for the Schrödinger operator under constant magnetic field

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