scholarly journals Wavelet analogue of the Ginzburg–Landau energy and its Γ-convergence

2010 ◽  
pp. 497-525 ◽  
Author(s):  
Julia Dobrosotskaya ◽  
Andrea Bertozzi
Keyword(s):  
Ginzburg Landau ◽  
Γ Convergence

scholarly journals Γ-convergence of a shearlet-based Ginzburg–Landau energy

2020 ◽  
Vol 49 (3) ◽  
pp. 727-770
Author(s):  
Philipp Christian Petersen ◽  
Endre Süli
Keyword(s):  
Ginzburg Landau ◽  
Γ Convergence

Relaxation of Ginzburg–Landau functional perturbed by continuous nonlinear lower-order term in one dimension

2014 ◽  
Vol 13 (01) ◽  
pp. 101-123 ◽  
Author(s):  
Andrija Raguž
Keyword(s):  
Asymptotic Behavior ◽  
Lower Order ◽  
Order Term ◽  
Lower Order Term ◽  
One Dimension ◽  
Young Measures ◽  
Ginzburg Landau ◽  
Carathéodory Function ◽  
Γ Convergence

We study the asymptotic behavior as ε → 0 of the Ginzburg–Landau functional [Formula: see text], 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 [Formula: see text] 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 [Formula: see text] is close to [Formula: see text], 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.

GINZBURG–LANDAU VORTEX LINES AND THE ELASTICA FUNCTIONAL

2009 ◽  
Vol 11 (01) ◽  
pp. 71-107
Author(s):  
ROGER MOSER
Keyword(s):  
Asymptotic Behavior ◽  
Second Order ◽  
Ginzburg Landau ◽  
Vortex Lines ◽  
Γ Convergence ◽  
Generalized Curves

We examine the asymptotic behavior of a family of second-order functionals arising in the theory of Ginzburg–Landau vortices. The results point towards Γ-convergence with the elastica functional for generalized curves as the limit.

Low energy configurations of topological singularities in two dimensions: A Γ-convergence analysis of dipoles

2019 ◽  
Vol 22 (03) ◽  
pp. 1950019
Author(s):  
Lucia De Luca ◽  
Marcello Ponsiglione
Keyword(s):  
Convergence Analysis ◽  
Two Dimensions ◽  
Low Energy ◽  
Ginzburg Landau ◽  
Energy Plus ◽  
Γ Convergence ◽  
Flat Norm ◽  
Topological Singularities ◽  
Energy Configurations ◽  
Small Zero

This paper deals with the variational analysis of topological singularities in two dimensions. We consider two canonical zero-temperature models: the core radius approach and the Ginzburg–Landau energy. Denoting by [Formula: see text] the length scale parameter in such models, we focus on the [Formula: see text] energy regime. It is well known that, for configurations whose energy is bounded by [Formula: see text], the vorticity measures can be decoupled into the sum of a finite number of Dirac masses, each one of them carrying [Formula: see text] energy, plus a measure supported on small zero-average sets. Loosely speaking, on such sets the vorticity measure is close, with respect to the flat norm, to zero-average clusters of positive and negative masses. Here, we perform a compactness and [Formula: see text]-convergence analysis accounting also for the presence of such clusters of dipoles (on the range scale [Formula: see text], for [Formula: see text]), which vanish in the flat convergence and whose energy contribution has, so far, been neglected. Our results refine and contain as a particular case the classical [Formula: see text]-convergence analysis for vortices, extending it also to low energy configurations consisting of just clusters of dipoles, and whose energy is of order [Formula: see text] with [Formula: see text].

scholarly journals Large data limit for a phase transition model with the p-Laplacian on point clouds

2018 ◽  
Vol 31 (2) ◽  
pp. 185-231 ◽  
Author(s):  
R. CRISTOFERI ◽  
M. THORPE
Keyword(s):  
Phase Transition ◽  
Large Data ◽  
Point Clouds ◽  
Mathematical Tool ◽  
Ginzburg Landau ◽  
Data Points ◽  
Non Local ◽  
Γ Convergence ◽  
Double Well Potential ◽  
Objective Functional

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.

scholarly journals Γ-Convergence of 2D Ginzburg-Landau functionals with vortex concentration along curves

2011 ◽  
Vol 114 (1) ◽  
pp. 341-391 ◽  
Author(s):  
Sam Alama ◽  
Lia Bronsard ◽  
Vincent Millot
Keyword(s):  
Ginzburg Landau ◽  
Γ Convergence

scholarly journals Asymptotic analysis of the Ginzburg–Landau functional on point clouds

2018 ◽  
Vol 149 (2) ◽  
pp. 387-427 ◽  
Author(s):  
Matthew Thorpe ◽  
Florian Theil
Keyword(s):  
Point Clouds ◽  
Transition Model ◽  
Interaction Potentials ◽  
Total Variation Norm ◽  
Ginzburg Landau ◽  
Data Points ◽  
Variation Norm ◽  
Γ Convergence ◽  
Anisotropic Case ◽  
Classification Type

AbstractThe 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 Ψnwherendenotes 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.

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