Optimal uniform elliptic estimates for the Ginzburg-Landau system

Fournais, Søren; Helffer, Bernard

doi:10.1090/conm/447/08684

Cited by 10 publications

(15 citation statements)

References 23 publications

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“…The result we need for proving the proposition is then that if −∆ψ is in addition in W k,p (Ω) then ψ ∈ W k+2,p (Ω). This is simply an L p regularity result for the Dirichlet problem for the Laplacian which is described in ( [6], Section F.4).…”

Section: 3mentioning

confidence: 98%

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The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

Attar

2015

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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Abstract. We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. We determine an accurate asymptotic formula for the minimizing energy when the Ginzburg-Landau parameter and the magnetic field are large and of the same order. As a consequence, it is shown how bulk superconductivity decreases in average as the applied magnetic field increases.

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Section: 3mentioning

confidence: 98%

“…This is simply the Dirichlet L p problem for the Laplacian (See [6], Section A.1). The result we need for proving the proposition is then that if −∆ψ is in addition in W k,p (Ω) then ψ ∈ W k+2,p (Ω).…”

Section: 3mentioning

confidence: 99%

The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

Attar

2015

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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“…In particular, we avoid the use of the elliptic estimates obtained by 'blow-up' techniques (cf. [Pan2] and see also [LuPa1,Alm,FoHe4]). …”

Section: The Analysis Of H C3mentioning

confidence: 99%

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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“…We first show intermediate lemmas. The first one recalls some useful a priori estimates (see [FH2,Proposition 4.4…”

Section: Reduction To a Torusmentioning

confidence: 99%

“…Fournais-Helffer [FH2]). The behavior of u, on the contrary, is strongly dependent on b: when b is too large (b > b 0 1.695), the only critical point of the energy is the normal solution u = 0, and A is such that curl A = h ex (see [GP]).…”

Section: Introductionmentioning

confidence: 99%

Lowest Landau level approach in superconductivity for the Abrikosov lattice close to $$H_{{c}_{2}}$$

Aftalion

Serfaty

2007

Sel. math., New ser.

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We study the Ginzburg-Landau energy for a superconductor submitted to a magnetic field h ex just below the "second critical field" H c 2 . When the Ginzburg-Landau parameter ε is small, we show that the mean energy per unit volume can be approximated by a reduced energy on a torus. Moreover, we expand this reduced energy in terms of H c 2 − h ex : when this quantity gets small, the problem amounts to a minimization problem on a finite-dimensional space, equivalent to the "lowest Landau level" in other approaches. The functions in this finite-dimensional space can themselves be expressed via the Jacobi Theta function of a lattice. This connects the Ginzburg-Landau energy to the "Abrikosov problem" of locating vortices optimally on a lattice. Mathematics Subject Classification (2000). 82D55, 35B40, 35B25, 35J60, 35J20, 58E50.

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Optimal uniform elliptic estimates for the Ginzburg-Landau system

Abstract: Abstract. We reconsider the elliptic estimates for magnetic operators in two and three dimensions used in connection with Ginzburg-Landau theory. Furthermore we discuss the so-called blow-up technique in order to obtain optimal estimates in the limiting cases.

Cited by 10 publications

References 23 publications

The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

The ground state energy of the two dimensional Ginzburg–Landau functional with variable magnetic field

On the Ginzburg‐Landau critical field in three dimensions

Lowest Landau level approach in superconductivity for the Abrikosov lattice close to $$H_{{c}_{2}}$$

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