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We consider a gauge-invariant Ginzburg–Landau functional (also known as Abelian Yang–Mills–Higgs model), on Hermitian line bundles over closed Riemannian manifolds of dimension n ≥ 3 {n\geq 3} . Assuming a logarithmic energy bound in the coupling parameter, we study the asymptotic behaviour of critical points in the London limit. After a convenient choice of the gauge, we show compactness of finite-energy critical points in Sobolev norms. Moreover, thanks to a suitable monotonicity formula, we prove that the energy densities of critical points, rescaled by the logarithm of the coupling parameter, converge to the weight measure of a stationary, rectifiable varifold of codimension 2. [ABSTRACT FROM AUTHOR]

Image inpainting models and the corresponding numerical algorithms play key roles in image processing. At present, the visual output of the oscillatory inpainting area is usually not natural. In this paper, we propose an image inpainting model based on the Ginzburg-Landau functional and H - 1 -norm. In the model, the H - 1 -fidelity term performs well in preserving the edges of the oscillatory inpainting areas, and the Ginzburg-Landau functional can provide additional geometric content. Theoretically, we prove the existence of the minimizer for the proposed energy functional. Based on the scalar auxiliary variable approach, we develop an efficient numerical scheme to solve the proposed model. Further, we use a time step adaptive strategy to accelerate the convergence. Experimental results validate the effectiveness of the proposed algorithm for image inpainting. [ABSTRACT FROM AUTHOR]

Tetrahedral frame fields have applications to certain classes of nematic liquid crystals and frustrated media. We consider the problem of constructing a tetrahedral frame field in three-dimensional domains in which the boundary normal vector is included in the frame on the boundary. To do this, we identify an isomorphism between a given tetrahedral frame and a symmetric, traceless third-order tensor under a particular nonlinear constraint. We then define a Ginzburg–Landau-type functional which penalizes the associated nonlinear constraint. Using gradient descent, one retrieves a globally defined limiting tensor outside of a singular set. The tetrahedral frame can then be recovered from this tensor by a determinant maximization method, developed in this work. The resulting numerically generated frame fields are smooth outside of one-dimensional filaments that join together at triple junctions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the minimization problem related to a Ginzburg–Landau energy functional, where in particular a nonlinear diffusion of mean curvature-type is considered, together with a classical double well potential. A careful analysis of the corresponding Euler–Lagrange equation, equipped with natural boundary conditions and mass constraint, leads to the existence of an unique Maxwell solution , namely a monotone increasing solution obtained for small diffusion and close to the so-called Maxwell point. Then, it is shown that this particular solution (and its reversal) has least energy among all the stationary points satisfying the given mass constraint. Moreover, as the viscosity parameter tends to zero, it converges to the increasing (decreasing for the reversal) single interface solution , namely the constrained minimizer of the corresponding energy without diffusion. Connections with Cahn–Hilliard models, obtained in terms of variational derivatives of the total free energy considered here, are also presented. [ABSTRACT FROM AUTHOR]

We present two graph-based algorithms for multiclass segmentation of high-dimensional data on graphs. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation and graph cuts. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, image labeling, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art in multiclass graph-based segmentation algorithms for high-dimensional data.

We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg-Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman-Bence-Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples. © 2014 Elsevier Ltd. All rights reserved.

In the continuum, close connections exist between mean curvature flow, the Allen-Cahn (AC) partial differential equation, and the Merriman-Bence-Osher (MBO) threshold dynamics scheme. Graph analogues of these processes have recently seen a rise in popularity as relaxations of NP-complete combinatorial problems, which demands deeper theoretical underpinnings of the graph processes. The aim of this paper is to introduce these graph processes in the light of their continuum counterparts, provide some background, prove the first results connecting them, illustrate these processes with examples and identify open questions for future study. We derive a graph curvature from the graph cut function, the natural graph counterpart of total variation (perimeter). This derivation and the resulting curvature definition differ from those in earlier literature, where the continuum mean curvature is simply discretized, and bears many similarities to the continuum nonlocal curvature or nonlocal means formulation. This new graph curvature is not only relevant for graph MBO dynamics, but also appears in the variational formulation of a discrete time graph mean curvature flow. We prove estimates showing that the dynamics are trivial for both MBO and AC evolutions if the parameters (the time-step and diffuse interface scale, respectively) are sufficiently small (a phenomenon known as "freezing" or "pinning") and also that the dynamics for MBO are nontrivial if the time step is large enough. These bounds are in terms of graph quantities such as the spectrum of the graph Laplacian and the graph curvature. Adapting a Lyapunov functional for the continuum MBO scheme to graphs, we prove that the graph MBO scheme converges to a stationary state in a finite number of iterations. Variations on this scheme have recently become popular in the literature as ways to minimize (continuum) nonlocal total variation. © 2014 The Author(s).

In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090--1118], Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg--Landau functional for solving problems in data classification. Here, we propose an adaptation of the classic numerical Merriman--Bence--Osher (MBO) scheme for minimizing graph-based diffuse interface functionals, like those originally proposed by Bertozzi and Flenner. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. Various computational examples are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. The results show that our method is multiple times more efficient than other well-known nonlocal models. © 2013 Society for Industrial and Applied Mathematics.

The consistency of a non-local anisotropic Ginzburg–Landau type functional for data classification and clustering is studied. The Ginzburg–Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter. [ABSTRACT FROM AUTHOR]

The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψ n where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence. [ABSTRACT FROM AUTHOR]

Abstract For n ≥ 3 and 0 < ϵ ≤ 1 , let Ω ⊂ R n be a bounded, simply connected, smooth domain, and u ϵ : Ω ⊂ R n → R 2 solve the Ginzburg–Landau equation under the weak anchoring boundary condition: { − Δ u ϵ = 1 ϵ 2 (1 − | u ϵ | 2) u ϵ in Ω , ∂ u ϵ ∂ ν + λ ϵ (u ϵ − g ϵ) = 0 on ∂ Ω , where the anchoring strength parameter λ ϵ = K ϵ − α for some K > 0 and α ∈ [ 0 , 1) , and g ϵ ∈ C 2 (∂ Ω , S 1). Motivated by the connection with the Landau–De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the asymptotic behavior of u ϵ as ϵ goes to zero under the condition that the total modified Ginzburg–Landau energy satisfies F ϵ (u ϵ , Ω) ≤ M | log ⁡ ϵ | for some M > 0. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of minimizing solutions of a Ginzburg–Landau type functional with a positive weight and with convex potential near 0 and we estimate the energy in this case. We also generalize a lower bound for the energy of unit vector field given initially by Brezis–Merle–Rivière.

We compute the L2-norm of any minimizer of the Ginzburg-Landau functional in a planar domain with a finite number of corners. Our computations are valid for a uniform applied magnetic field, large Ginzburg-Landau parameter and in the regime where superconductivity is confined near the corners of the domain. [ABSTRACT FROM AUTHOR]

The Landau–de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state. [ABSTRACT FROM AUTHOR]

The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface. [ABSTRACT FROM AUTHOR]

We explore the relationship between two reference functions arising in the analysis of the Ginzburg–Landau functional. The first function describes the distribution of superconductivity in a type II superconductor subjected to a constant magnetic field. The second function describes the distribution of superconductivity in a type II superconductor submitted to a variable magnetic field that vanishes non-degenerately along a smooth curve. [ABSTRACT FROM AUTHOR]

We study the existence of solutions for a Ginzburg-Landau ODE in the half-axis. We obtain that the solution is unique in the class of non-negative finite energy solutions. As a by-product, we determine the minimizers of the corresponding Ginzburg-Landau functional. [ABSTRACT FROM AUTHOR]

We study compactness and $Gamma$-convergence for Ginzburg-Landau type functionals. We only assume that the potential is continuous and positive definite close to one circular well, but allow large zero sets inside the well. We show that the relaxation of the assumptions does not change the results to leading order unless the energy is very large.

We have developed a numerical method for approximating critical points of the Ginzburg-Landau functional. We briefly describe the capabilities of our codes and present some test results for the two-dimensional case in the form of plots of electron densities and magnetic fields. Our results demonstrate that vortices can be pinned to small holes corresponding to normal (non-superconducting) regions.

We study the asymptotic behavior of the regularizable minimizers of a Ginzburg-Landau type functional. We also dicuss the location of the zeroes of the minimizers.

In this article we study the minimizers of the functional $$ E_varepsilon(u,G)={1over p}int_G|abla u|^p+frac{1 over 4varepsilon^p} int_G(1-|u|^2)^2, $$ on the class $W_g={v in W^{1,p}(G,{mathbb R}^2);v|_{partial G}=g}$, where $g:partial G o S^1$ is a smooth map with Brouwer degree zero, and $p$ is greater than 2. In particular, we show that the minimizer converges to the $p$-harmonic map in $C_{hbox{loc}}^{1,alpha}(G,{mathbb R}^2)$ as $varepsilon$ approaches zero.

The minimization of volume constrained vector-valued Ginzburg–Landau energy functional is considered in the present study. It has many applications in computational science and engineering, like the conservative phase separation in multiphase systems (such as the spinodal decomposition), phase coarsening in multiphase systems, color image segmentation and optimal space partitioning. A computationally efficient algorithm is presented to solve the space discretized form of the original optimization problem. The algorithm is based on the constrained nonmonotone L 2 gradient flow of Ginzburg–Landau functional followed by a regularization step, which is resulted from the Tikhonov regularization term added to the objective functional, that lifts the solution from the L 2 function space into H 1 space. The regularization step not only improves the convergence rate of the presented algorithm, but also increases its stability bound. The step-size selection based on the Barzilai–Borwein approach is adapted to improve the convergence rate of the introduced algorithm. The success and performance of the presented approach is demonstrated throughout several numerical experiments. To make it possible to reproduce the results presented in this work, the MATLAB implementation of the presented algorithm is provided as the supplementary material. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior as ε → 0 of the Ginzburg-Landau functional , where A(s, v, v′) is the nonlinear lower-order term generated by certain Carathéodory function a : (0, 1)2 × R2 → R. We obtain Γ-convergence for the rescaled functionals as ε → 0 by using the notion of Young measures on micropatterns, which was introduced in 2001 by Alberti and Müller. We prove that for ε ≈ 0 the minimal value of is close to , where A∞(s) : = ½A(s, 0, -1) + ½A(s, 0, 1) and where E0 depends only on W. Further, we use this example to establish some general conclusions related to the approach of Alberti and Müller. [ABSTRACT FROM AUTHOR]

We consider the functional $$ E_varepsilon(u,G) =frac 1pint_G|abla u|^p +frac{1}{4varepsilon^p}int_G(1-|u|^2)^2 $$ with $p>2$ and $d>0$, on the class of functions $W={u(x)=f(r)e^{idheta} in W^{1,p}(B,C); f(1)=1,f(r)geq 0}$. The location of the zeroes of the minimizer and its convergence as $varepsilon$ approaches zero are established.

The human and scientific personality of M. K. Venkatesha Murthy is illustrated and his great spirit of cooperation and solidarity is highlighted. [ABSTRACT FROM AUTHOR]

Abstract: Sobolev gradients have been discussed in Sial et al. (2003) as a method for energy minimization related to Ginzburg–Landau functionals. In this article, a weighted Sobolev gradient approach for the time evolution of a Ginzburg–Landau functional is presented for different values of . A comparison is given between the weighted and unweighted Sobolev gradients in a finite element setting. It is seen that for small values of , the weighted Sobolev gradient method becomes more and more efficient compared to using the unweighted Sobolev gradient. A comparison with Newton’s method is given where the failure of Newton’s method is demonstrated for a test problem. [Copyright &y& Elsevier]

Abstract: We study the Ginzburg–Landau model of superconductivity in three dimensions and for strong external magnetic fields. For magnetic field strengths above the phenomenologically defined second critical field it is known from Physics that superconductivity should be essentially restricted to a region near the boundary. We prove that the expected region does indeed carry superconductivity. Furthermore, we give precise energy estimates valid also in the regime around the second critical field which display the transition from bulk superconductivity to surface superconductivity. [Copyright &y& Elsevier]

We consider a homogenization problem for the magnetic Ginzburg-Landau functional in domains with a large number of small holes. We establish a scaling relation between sizes of holes and the magnitude of the external magnetic field when the multiple vortices pinned by holes appear in nested subdomains and their homogenized density is described by a hierarchy of variational problems. This stands in sharp contrast with homogeneous superconductors, where all vortices are known to be simple. The proof is based on the Γ-convergence approach applied to a coupled continuum/discrete variational problem: continuum in the induced magnetic field and discrete in the unknown finite (quantized) values of multiplicity of vortices pinned by holes. [ABSTRACT FROM AUTHOR]

We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the magnetic field is comparable with the second critical field and that the Ginzburg-Landau parameter is large, we determine a precise asymptotic formula for the minimizing energy. In particular, this shows how bulk superconductivity decreases in average as the applied magnetic field approaches the second critical field from below. Other estimates are also obtained which allow us to obtain, in a subsequent paper [19], a fine characterization of the second critical field. The approach relies on a careful analysis of several limiting energies, which is of independent interest. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the pinning effect for full Ginzburg-Landau functional on a superconducting hollow sphere. The existence of local minimizers with vortices locating in the pinning regions is obtained. [ABSTRACT FROM AUTHOR]

Abstract: We consider the Ginzburg–Landau functional defined over a bounded and smooth three-dimensional domain. Supposing that the strength of the applied magnetic field varies between the first and second critical fields, in such a way that , we estimate the ground state energy to leading order as the Ginzburg–Landau parameter tends to infinity. [Copyright &y& Elsevier]

We consider minimizers of the Ginzburg-Landau energy with pinning term and zero degree Dirichlet boundary condition. Without any assumptions on the pinning term, we prove that these minimizers do not develop vortices in the limit ε → 0. We next consider the specific case of a periodic discontinuous pinning term taking two values. In this setting, we determine the asymptotic behavior of the minimizers as ε → 0. [ABSTRACT FROM AUTHOR]

In the presence of applied magnetic fields H in the order of H Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. the first critical field, we determine the limiting vorticities of the minimal Ginzburg-Landau energy with weight. Our result is analogous to the work of E. Sandier and S. Serfaty [26]. [ABSTRACT FROM AUTHOR]

In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schrödinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to infinity. This asymptotic formula is used to prove that the eigenvalue is monotonically increasing for large values of the magnetic field. [ABSTRACT FROM AUTHOR]

Abstract: We consider the two-dimensional Ginzburg–Landau functional with constant applied magnetic field. For applied magnetic fields close to the second critical field and large Ginzburg–Landau parameter, we provide leading order estimates on the energy of minimizing configurations. We obtain a fine threshold value of the applied magnetic field for which bulk superconductivity contributes to the leading order of the energy. Furthermore, the energy of the bulk is related to that of the Abrikosov problem in a periodic lattice. A key ingredient of the proof is a novel -bound which is of independent interest. [Copyright &y& Elsevier]

In the presence of applied magnetic fields H such that | ln ɛ| ≪ H ≪ $$ \tfrac{1} {{\varepsilon ^2 }} $$, the author evaluates the minimal Ginzburg-Landau energy with discontinuous constraint. Its expression is analogous to the work of Sandier and Serfaty. [ABSTRACT FROM AUTHOR]

We study compactness and Γ-convergence for Ginzburg-Landau type functionals. We only assume that the potential is continuous and positive definite close to one circular well, but allow large zero sets inside the well. We show that the relaxation of the assumptions does not change the results to leading order unless the energy is very large. [ABSTRACT FROM AUTHOR]

Abstract: We consider, in a smooth bounded multiply connected domain , the Ginzburg–Landau energy subject to prescribed degree conditions on each component of . In general, minimal energy maps do not exist [L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers in perforated domains with prescribed degrees, preprint, 2004]. When 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]: has, in domains with 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. [Copyright &y& Elsevier]

Abstract: Although the Poincaré and the geometrization conjectures were recently proved by Perelman, the proof relies heavily on properties of the Ricci flow previously investigated in great detail by Hamilton. Physical realization of such a flow can be found, for instance, in the work by Friedan [D. Friedan, Nonlinear models in dimensions, Ann. Phys. 163 (1985) 318–419]. In his work the renormalization group flow for a nonlinear sigma model in dimensions was obtained and studied. For , by approximating the -function for such a flow by the lowest order terms in the sigma model coupling constant, the equations for Ricci flow are obtained. In view of such an approximation, the existence of this type of flow in Nature is questionable. In this work, we find totally independent justification for the existence of Ricci flows in Nature. This is achieved by developing a new formalism extending the results of two-dimensional conformal field theories (CFT’s) to three and higher dimensions. Equations describing critical dynamics of these CFT’s are examples of the Yamabe and Ricci flows realizable in Nature. Although in the original works by Perelman some physically motivated arguments can be found, their role in his proof remain rather obscure. In this paper, steps are made toward making these arguments more explicit, thus creating an opportunity for developing alternative, more physically motivated, proofs of the Poincaré and geometrization conjectures. [Copyright &y& Elsevier]