Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data

We are interested in proving Monte-Carlo approximations for 2d Navier-Stokes equations with initial data u ₀ belonging to the Lorentz space L ^2,∞ and such that curl u ₀ is a finite measure. Giga, Miyakawaand Osada [7] proved that a solution u exists and that u=K* curl u, where K is the Biot-Savartkernel and v = curl u is solution of a nonlinear equation in dimension one, called the vortex equation. In this paper, we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a Monte-Carlo approximation for a solution of the Navier-Stokesequation. That gives in this case a pathwise proofof the vortex algorithm introducedby Chorin and consequently generalizes the works ofMarchioro-Pulvirenti [12] and Méléardv [15] obtained in the case of a vortex equation with bounded density initial data. Received: 6 October 1999 / Revised version: 15 September 2000 / Published online: 9 October 2001     

Benchmarking optimization software with performance profiles

We propose performance profiles — distribution functions for a performance metric — as a tool for benchmarking and comparing optimization software. We show that performance profiles combine the best features of other tools for performance evaluation. Received: February 2001 / Accepted: May 2001?Published online October 2, 2001     

A Bernstein theorem for special Lagrangian graphs

We obtain a Bernstein theorem for special Lagrangian graphs in for arbitrary n only assuming bounded slope but no quantitative restriction. Received: 18 January 2001 / Accepted: 7 June 2001 / Published online: 12 October 2001 The second-named author is grateful to the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality and support and also 973 project in China.     

Renormalized energy of interacting Ginzburg-Landau vortex filaments

We consider the Ginzbug–Landau energy in a cylinder in³, and a canonical approximation for critical points with anassembly of n2 periodic vortex lines near the axis of the cylinder.We find a formula for the energy which, up to a large additiveconstant and to leading order, is the action functional of then-body problem with a logarithmic potential in ², the axis variableplaying the role of time. A special family of rotating helicoidalcritical points of the functional is found to be non-degenerateup to the invariances of the problem, and therefore persistentunder small perturbations. Our analysis suggests the presenceof very complex stationary configurations for vortex filaments,potentially also involving intersecting filaments.     

A partial regularity result for harmonic maps into a Finsler manifold

In this paper, we consider the energy of maps from an Euclidean space into a Finsler space and study the partial regularity of energy minimizing maps. We show that the -dimensional Hausdorff measure of the singular set of every energy minimizing map is 0 for some , when m=3,4. Received: 6 June 2001 / Accepted: 10 July 2001 / Published online: 12 October 2001     

Static Vortex Solutions in a Planar Particle System Obeying a Generalized Exclusion Principle

We consider a planar particle system obeying a generalized Pauli exclusion principle. In the mean field approximation, this system is described by a Schrödinger equation we recently introduced, containing a complex nonlinearity. The particle number, the total energy, and the angular momentum are conserved in such a system. We consider vortexlike stationary solutions of the form and write the differential equation for the vortex shape. We find an analytic solution of this equation and obtain a closed expression for the vortex profile. We investigate some mean properties and, in particular, calculate the energy spectrum and angular momentum of the vortex.     

A Gibbs variational principle in space-time for infinite-dimensional diffusions

 We show that the set of stationary weak solutions for a class of infinite dimensional stochastic differential equations coincides with the set of shift invariant, space-time Gibbs fields for a certain potential. The key step consists in proving the Gibbs variational principle for space-time Gibbs fields. Received: 20 May 1999 / Revised version: 14 May 2001 / Published online: 11 December 2001     

Vortex energy and 360° Néel walls in thin‐film micromagnetics

We study the vortex pattern in ultrathin ferromagnetic films of circular crosssection. The model is based on the following energy functional: for in‐plane magnetizations m: B² → S¹ in the unit disc . The avoidance of volume charges ? · m ≠ 0 in B² and surface charges m · ν ≠ 0 on δB² leads to the formation of a vortex in the limit ε → 0. At the level ε > 0 the vortex is regularized by the formation of a 360° Néel wall (a one‐dimensional transition layer with core of scale ε) concentrated along a radius of B². We derive the limiting energy of the vortex by matching upper and lower bounds. Our analysis on the lower bound is based on a dynamical system argument and an interpolation inequality with sharp leading‐order constant, while the upper bound uses the leading‐order energy for 360° Néel walls. © 2010 Wiley Periodicals, Inc.     

A unimodularly invariant theory for immersions into the affine space

We develop a unimodularly invariant theory for immersions with higher codimension into the affine space. Received: 6 September 2001; in final form: 22 November 2001 / Published online: 29 April 2002 RID="*" ID="*" Supported by the Deutsche Forschungsgemeinschaft     

A dynamical systems model of unorganized segregation

We consider Schelling’s bounded neighborhood model (BNM) of unorganized segregation, from the perspective of modern dynamical systems theory. We carry out a complete quantitative analysis of the system for linear tolerance schedules. We derive a fully predictive model and associate each term with a social meaning. We recover and generalize Schelling’s qualitative results.

For the case of unlimited population movement, we derive exact formulae for regions in parameter space where stable integrated population mixes can occur, and show how neighborhood tipping can be explained in terms of basins of attraction.

When population movement is limited, we derive exact criteria for the occurrence of new population mixes.

For nonlinear tolerance schedules, we illustrate our approach with numerical simulations.     

A quantization property for static Ginzburg‐Landau vortices

For any critical point of the complex Ginzburg‐Landau functional in dimension 3, we prove that, for large coupling constants, ; if the energy of this critical point on a ball of a given radius r is relatively small compared to r log , then the ball of half‐radius contains no vortex (the modulus of the solution is larger than ½). We then show how this property can be applied to describe limiting vortices as ε → 0. © 2001 John Wiley & Sons, Inc.     

Algebraic models for equivariant homotopy theory over Abelian compact Lie groups

We describe some algebraic models for equivariant rational and p-adic homotopy theory over Abelian compact Lie groups. Received: 12 February 2001; in final form: 15 August 2001 / Published online: 28 February 2002     

Hypersurfaces of prescribed Gauß curvature in exterior domains

We prove an existence theorem for convex hypersurfaces of prescribed Gau? curvature in the complement of a compact set in Euclidean space which are close to a cone. Received: 23 February 2001 / Accepted: 11 May 2001 / Published online: 19 October 2001     

Twodimensional anisotropic variational problems

We consider local minimizers a domain in , of the variational integral with integrand f of upper (lower) growth rate q (s). We show using a lemma due to Frehse and Seregin that u has H?lder continuous first derivatives provided that . Received: 2 October 2001 / Accepted: 25 October 2001 / Published online: 28 February 2002     

Integration of large-deviation kernels and applications to large deviations for evolutionary games

 We investigate large-deviation properties for systems of probability kernels and these kernels when integrated with respect to a system of measures that also satisfy the large-deviation principle. The results of our analysis are then used to demonstrate the large-deviation property and derive rate functions for systems of measures derived from the distributions of evolutionary game processes. Received: 26 October 1998 / Revised version: 17 April 2001 / Published online: 11 December 2001     

Once edge-reinforced random walk on a tree

 We consider a nearest neighbor walk on a regular tree, with transition probabilities proportional to weights or conductances of the edges. Initially all edges have weight 1, and the weight of an edge is increased to $c > 1$ when the edge is traversed for the first time. After such a change the weight of an edge stays at $c$ forever. We show that such a walk is transient for all values of $c \ge 1$, and that the walk moves off to infinity at a linear rate. We also prove an invariance principle for the height of the walk. Received: 6 March 2001 / Revised version: 16 July 2001 / Published online: 15 March 2002     

The doubly periodic Costa surfaces

We derive global Weierstrass representations for complete minimal surfaces obtained by substituting the planar end of the Costa surface by symmetry curves. Received: 14 February 2001; in final form: 24 April 2001 / Published online: 29 April 2002     

A topological aspect of Sobolev mappings

Let be a connected open set, . We give a sufficient condition for a mapping , , to have the property that sgn is almost everywhere of one sign. Following the work of Müller, Spector, and Tang [MST], we give an application of our results to the theory of non-linear elasticity. Received: 13 October 2000 / Accepted: 23 January 2001 / Published online: 4 May 2001