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1.
We study the variational convergence of a family of twodimensional Ginzburg-Landau functionals arising in the study of superfluidity
or thin-film superconductivity as the Ginzburg-Landau parameter ε tends to 0. In this regime and for large enough applied rotations (for superfluids) or magnetic fields (for superconductors),
the minimizers acquire quantized point singularities (vortices). We focus on situations in which an unbounded number of vortices
accumulate along a prescribed Jordan curve or a simple arc in the domain. This is known to occur in a circular annulus under
uniform rotation, or in a simply connected domain with an appropriately chosen rotational vector field. We prove that if suitably
normalized, the energy functionals Γ-converge to a classical energy from potential theory. Applied to global minimizers, our
results describe the limiting distribution of vortices along the curve in terms of Green equilibrium measures. 相似文献
2.
Complex Ginzburg-Landau equations and dynamics of vortices,filaments, and codimension-2 submanifolds
Fang Hua Lin 《纯数学与应用数学通讯》1998,51(4):385-441
Here we study the asymptotic behavior of solutions to the complex Ginzburg-Landau equations and their associated heat flows in arbitrary dimensions when the Ginzburg-Landau parameter tends to infinity. We prove that the energies of solutions in the flow are concentrated at vortices in two dimensions, filaments in three dimensions, and codimension-2 submanifolds in higher dimensions. Moreover, we show the dynamical laws for the motion of these vortices, filaments, and codimension-2 submanifolds. Away from the energy concentration sets, we use some measure-theoretic arguments to show the strong convergence of solutions in both static and heat flow cases. © 1998 John Wiley & Sons, Inc. 相似文献
3.
Yan Guo 《纯数学与应用数学通讯》1996,49(10):1051-1080
In the Ginzburg-Landau theory for superconductivity, it is well-known that there are radial symmetric vortices for any charge (vortex number) n and coupling constant λ > 0. It is powered that these vortices are unstable for large n and λ. © 1996 John Wiley & Sons, Inc. 相似文献
4.
In this paper we prove that Ginzburg-Landau vortices with charge n, |n| > 1, and a large coupling constant are dynamically unstable (using the Maxwell-Higgs model). © 1997 John Wiley & Sons, Inc. 相似文献
5.
Kou Yanlei Ding Shijin 《高校应用数学学报(英文版)》2007,22(1):48-60
In this paper, it is proved that for any given d non-degenerate local minimum points of the renormalized energy of weighted Ginzburg-Landau eqautions, one can find solutions to the Ginzburg-Landau equations whose vortices tend to these d points. This provides the connections between solutions of a class of Ginzburg-Landau equations with weight and minimizers of the renormalized energy. 相似文献
6.
A. G. Sergeev 《Proceedings of the Steklov Institute of Mathematics》2010,270(1):230-239
We study an adiabatic limit in (2 + 1)-dimensional hyperbolic Ginzburg-Landau equations and 4-dimensional symplectic Seiberg-Witten
equations. In dimension 3 = 2+1 the limiting procedure establishes a correspondence between solutions of Ginzburg-Landau equations
and adiabatic paths in the moduli space of static solutions, called vortices. The 4-dimensional adiabatic limit may be considered
as a complexification of the (2+1)-dimensional procedure with time variable being “complexified.” The adiabatic limit in dimension
4 = 2+2 establishes a correspondence between solutions of Seiberg-Witten equations and pseudoholomorphic paths in the moduli
space of vortices. 相似文献
7.
In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices. 相似文献
8.
Mickaël Dos Santos 《Journal of Functional Analysis》2009,257(4):1053-1762
We consider, in a smooth bounded multiply connected domain D⊂R2, the Ginzburg-Landau energy subject to prescribed degree conditions on each component of ∂D. In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg-Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When D has a single hole, Berlyand and Rybalko [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html] proved that for small ε local minimizers do exist. We extend the result in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]: Eε(u) has, in domains D with 2,3,… holes and for small ε, local minimizers. Our approach is very similar to the one in [L. Berlyand, V. Rybalko, Solution with vortices of a semi-stiff boundary value problem for the Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS), in press, 2008, http://www.math.psu.edu/berlyand/publications/publications.html]; the main difference stems in the construction of test functions with energy control. 相似文献
9.
In this paper we prove novel lower bounds for the Ginzburg-Landau energy with or without magnetic field. These bounds rely on an improvement of the “vortex-balls construction” estimates by extracting a new positive term in the energy lower bounds. This extra term can be conveniently estimated through a Lorentz space norm, on which it thus provides an upper bound. The Lorentz space L2,∞ we use is critical with respect to the expected vortex profiles and can serve to estimate the total number of vortices and get improved convergence results. 相似文献
10.
Given a bounded doubly connected domain G⊂R2, we consider a minimization problem for the Ginzburg-Landau energy functional when the order parameter is constrained to take S1-values on ∂G and have degrees zero and one on the inner and outer connected components of ∂G, correspondingly. We show that minimizers always exist for 0<λ<1 and never exist for λ?1, where λ is the coupling constant ( is the Ginzburg-Landau parameter). When λ→1−0 minimizers develop vortices located near the boundary, this results in the limiting currents with δ-like singularities on the boundary. We identify the limiting positions of vortices (that correspond to the singularities of the limiting currents) by deriving tight upper and lower energy bounds. The key ingredient of our approach is the study of various terms in the Bogomol'nyi's representation of the energy functional. 相似文献
11.
R. V. Palvelev A. G. Sergeev 《Proceedings of the Steklov Institute of Mathematics》2012,277(1):191-205
We study the adiabatic limit in hyperbolic Ginzburg-Landau equations which are the Euler-Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg-Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg-Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author. 相似文献
12.
In this paper we study the asymptotic behaviors of the likelihood ratio criterion (TL(s)), Watson statistic (TW(s)) and Rao statistic (TR(s)) for testing H0s: μ
(a given subspace) against H1s: μ
, based on a sample of size n from a p-variate Langevin distribution Mp(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ−1. For the case when κ is unknown, it is shown that TR(s) TL(s) TW(s) in their powers up to the order κ−1. 相似文献
13.
Hassen Aydi Etienne Sandier 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2009,26(4):1223-1236
We study the vortices of energy minimizers in the London limit for the Ginzburg–Landau model with periodic boundary conditions. For applied fields well below the second critical field we are able to describe the location and number of vortices. Many of the results presented appeared in [H. Aydi, Doctoral Dissertation, Université Paris-XII, 2004], others are new. 相似文献
14.
Richard F. Bass Krzysztof Burdzy 《Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques》2001,37(6):627
For
, we consider Lft, the local time of space-time Brownian motion on the curve f. Let
be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of Lf1 over f in
is finite if α>1/2 and infinite if α<1/2. 相似文献
15.
Kwang Ik KIM Zu Han LIU 《数学学报(英文版)》2008,24(1):75-86
We study the minimizers of the Ginzburg-Landau model for variable thickness, superconducting, thin films with high k, placed in an applied magnetic field hex, when hex is of the order of the "first critical field", i.e. of the order of |lnε|. We obtain the asymptotic estimates of minimal energy and describe the possible locations of the vortices. 相似文献
16.
The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented. 相似文献
17.
We consider a Ginzburg-Landau functional for a complex vector order parameter Ψ=(ψ+,ψ−), whose minimizers exhibit vortices with half-integer degree. By studying the associated system of equations in R2 which describes the local structure of these vortices, we show some new and unconventional properties of these vortices. In particular, one component of the solution vanishes, but the other does not. We also prove the existence and uniqueness of equivariant entire solutions, and provide a second proof of uniqueness, valid for a large class of systems with variational structure. 相似文献
18.
We prove that for fields close enough to the first critical field, minimizers of the Ginzburg-Landau functional have a number
of vortices bounded independently from the Ginzburg-Landau parameter. This generalizes a result proved in [SS1] and shows
that locally minimizing solutions of the Ginzburg-Landau equation found in [S1, S3] are actually global minimizers. It also
gives a partial answer to a question raised by F. Bethuel and T. Rivière in [BR].
Received: 10 July 2002 / Accepted: 23 January 2002 / Published online: 5 September 2002 相似文献
19.
We survey some recent work concerning the asymptotic dynamics of vortices in the 2-dimensional parabolic Ginzburg-Landau equation, the interaction of vortices with the phase field and the limiting initial value problem for both vortices and phase. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
20.
Circe Mary Silva da Silva 《Historia Mathematica》1999,26(4):327
This article describes the strong influence of positivism on the teaching of mathematics in Brazil. The dissemination of positivism occurred in a very intensive way from 1870 to 1930, due mainly to the strong leadership of teachers at the military and engineering academies. From its firmly entrenched position in these institutions, the positivistic ideology affected the social, political, pedagogical, and ideological life in Brazil. Here, I identify the main representatives of positivism, who focused their research on Auguste Comte's concept of mathematics. They oriented curricula and programs according to Comte's principles as well as produced mathematics with a distinct positivist bent. Although a marked decline occurred after 1930, the positivistic phenomenon was not exhausted as a research topic, and, indeed, it still has not been entirely extinguished in Brazilian life. Copyright 1999 Academic Press.Este trabalho descreve a forte influência do positivismo no ensino da Matemática no Brasil. A difusão do positivismo aconteceu de forma muita intensa entre 1870 e 1930, devido principalmente a atuação dos docentes-militares, que mantinham uma liderança forte nas academias militares e de engenharia. Nestas instituições a ideologia positivista encontrou uma forte sustentação e pode, então, ter efeitos na vida social, polı́tica, pedagógica e ideológica brasileira. Identificamos os principais representantes do positivismo no cı́rculo acadêmico. Detectamos as primeiras manifestações da concepção de Matemática de Auguste Comte em livros-texto. Identificamos a orientação de currı́culos e programas segundo os preceitos de Comte e analisamos principalmente as obras de Matemática de autores positivistas. O declı́nio do positivismos depois de 1930 também é registrado. O fenômeno positivismo não foi esgotado como tema de pesquisa e tudo indica que ainda não se extinguiu completamente da vida brasileira. Copyright 1999 Academic Press.MSC 1991 subject classifications: 01A55, 01A70. 相似文献