Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain

Berlyand, Leonid; Golovaty, Dmitry; Rybalko, Volodymyr

doi:10.1016/j.crma.2006.05.013

Cited by 14 publications

(45 citation statements)

References 10 publications

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“…In other words, we are interested in whether the model (1.1, 1.3) stabilizes vortices similarly to the Dirichlet problem or does not stabilize vortices analogously to the Neumann problem. The boundary conditions (1.3) are not well studied, and this work, along with studies [10,19,7,8,9], reveals their distinctive features, described later in the Introduction. Let us briefly review the existing results for the Dirichlet and Neumann boundary value problems for equation (1.1).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In [8], it was shown that minimizing sequences for the corresponding minimization problem develop a novel type of so-called "near-boundary" vortices, which approach the boundary and have finite GL energy in the limit of small ε. However, such minimizing sequences do not converge to actual minimizers [9]. These studies lead to the natural question of whether there exist true solutions of (1.1, 1.3) with near-boundary vortices.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Berlyand¹,

Rybalko

2010

J. Eur. Math. Soc.

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Abstract. We study solutions of the 2D Ginzburg-Landau equationsubject to "semi-stiff" boundary conditions: Dirichlet conditions for the modulus, |u| = 1, and homogeneous Neumann conditions for the phase. The principal result of this work shows that there are stable solutions of this problem with zeros (vortices), which are located near the boundary and have bounded energy in the limit of small ε. For the Dirichlet boundary condition ("stiff" problem), the existence of stable solutions with vortices, whose energy blows up as ε → 0, is well known. By contrast, stable solutions with vortices are not established in the case of the homogeneous Neumann ("soft") boundary condition.In this work, we develop a variational method which allows one to construct local minimizers of the corresponding Ginzburg-Landau energy functional. We introduce an approximate bulk degree as the key ingredient of this method; unlike the standard degree over the curve, it is preserved in the weak H 1 -limit.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Berlyand¹,

Rybalko

2010

J. Eur. Math. Soc.

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“…This result (in the case d = 1) was improved and extended in [6] to general doubly connected domains. The existence/nonexistence study for doubly connected domains was completed in [4]. Works [6] and [4] show that the existence of minimizers crucially depends on the upper energy bound obtained by minimizing the Dirichlet energy among S 1 -valued maps.…”

Section: Theoremmentioning

confidence: 99%

“…The existence/nonexistence study for doubly connected domains was completed in [4]. Works [6] and [4] show that the existence of minimizers crucially depends on the upper energy bound obtained by minimizing the Dirichlet energy among S 1 -valued maps. 3 Namely, if this bound does not exceed a certain threshold then minimizers exist for all ε, otherwise they exist for large ε and do not exist for small ε.…”

Section: Theoremmentioning

confidence: 99%

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Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

Berlyand

Mironescu

Rybalko

et al. 2014

Communications in Partial Differential Equations

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Let Ω ⊂ R 2 be a smooth bounded simply connected domain. We consider the simplified Ginzburg-2 , where u : Ω → C. We prescribe |u| = 1 and deg (u, ∂Ω) = 1. In this setting, there are no minimizers of E ε . Using a mountain pass approach, we obtain existence of critical points of E ε for large ε. Our analysis relies on Wente estimates and on the study of bubbling phenomena for Palais-Smale sequences.

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Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids

Hauswirth¹,

Rodiac

2016

Calc. Var.

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Let A ⊂ R 2 be a smooth doubly connected domain. We consider the Dirichlet energy E(u) = A |∇u| 2 , where u : A → C, and look for critical points of this energy with prescribed modulus |u| = 1 on ∂A and with prescribed degrees on the two connected components of ∂A. This variational problem is a problem with lack of compactness hence we can not use the direct methods of calculus of variations. Our analysis relies on the so-called Hopf differential and on a strong link between this problem and the problem of finding all minimal surfaces bounded by two p covering of circles in parallel planes. We then construct new immersed minimal surfaces in R 3 with this property. These surfaces are obtained by bifurcation from a family of p-coverings of catenoids.

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Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain

Cited by 14 publications

References 10 publications

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

Harmonic maps with prescribed degrees on the boundary of an annulus and bifurcation of catenoids

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