Logarithmic lower bounds for Néel walls

Most mathematical models for interfaces and transition layers in materials science exhibit sharply localized and rapidly decaying transition profiles. We show that this behavior can largely change when non-local interactions dominate and internal length scales fail to be determined by dimensional analysis: we consider a reduced model for Néel walls, micromagnetic transition layers which are observed in a suitable thin-film regime. The typical phenomenon associated with this wall type is the very long logarithmic tail of transition profiles. Recently, we derived logarithmic upper bounds. Here, we prove that the latter result is indeed optimal. In particular, we show that Néel wall profiles are supported by explicitly known comparison profiles that minimize relaxed variational principles and exhibit logarithmic decay behavior. This lower bound is established by a comparison argument based on a global maximum principle for the non-local field operator and the qualitative decay behavior of comparison profiles.Received: 17 June 2003, Accepted: 18 November 2003, Published online: 25 February 2004Mathematics Subject Classification (2000): 78A30, 49S05, 45G15, 35B25     

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

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.     

Differential forms in Carnot groups: a Γ-convergence approach

Carnot groups (connected simply connected nilpotent stratified Lie groups) can be endowed with a complex (E ₀ ^* , d _c ) of “intrinsic” differential forms. In this paper we prove that, in a free Carnot group of step κ, intrinsic 1-forms as well as their intrinsic differentials d _c appear naturally as limits of usual “Riemannian” differentials d _ε , ε >?0. More precisely, we show that L ²-energies associated with ε ^?κ d _ε on 1 forms Γ-converge, as ε → 0, to the energy associated with d _c .     

Une méthode de Γ-convergence pour un modèle de membrane non linéaire

We consider a cylindrical three-dimensional nonlinear hvperelastic body whose stored energy goes to infinity when the jacobian of the deformation gradient goes to zero. We show, under appropriate hypotheses on the applied loads, that the three-dimensional energies Γ-converge to the energy of a nonlinear membrane when the thickness of the cylinder goes to zero. The Γ-convergence takes place in the weak topology of a Sobolev space. As a consequence, we show that the sequence of minimizers also converges toward a minimizer of the limit energy.     

A novel approach towards fuzzy Γ-ideals in ordered Γ-semigroups

In many applied disciplines like computer science, coding theory and formal languages, the use of fuzzified algebraic structures especially ordered semigroups play a remarkable role. In this paper, we introduce a new concept of fuzzy Γ-ideal of an ordered Γ-semigroup G called an (∈, ∈ ?q _k )-fuzzy Γ-ideal of G. Fuzzy Γ-ideal of type (∈, ∈ ∨q _k ) are the generalization of ordinary fuzzy Γ-ideals of an ordered Γ-semigroup G. A new characterization of ordered Γ-semigroups in terms of an (∈, ∈ ∨q _k )-fuzzy Γ-ideal is given. We show that a fuzzy subset λ of an ordered Γ-semigroup G is an (∈, ∈ ∨q _k )-fuzzy Γ-ideal of G if and only if U (λ; t) is a Γ-ideal of G for all \(t \in \left( {0,\frac{{1 - k}} {2}} \right]\) . We also investigate some important characterization theorems in terms of this notion. Finally, regular ordered Γ-semigroups are characterized by the properties of their (∈, ∈ ∨q _k )-fuzzy Γ-ideals.     

2-d stability of the Néel wall

We are interested in thin-film samples in micromagnetism, where the magnetization m is a 2-d unit-length vector field. More precisely we are interested in transition layers which connect two opposite magnetizations, so called Néel walls.We prove stability of the 1-d transition layer under 2-d perturbations. This amounts to the investigation of the following singularly perturbed energy functional:

A regularity result for Calderón commutators

We present a regularity result for the Calderon commutator $[u,\mathcal H ](v)$ where $u,v$ are moduli of continuity and $\mathcal H $ is the Hilbert transform.     

A result on Waring–Goldbach problem for cubes

In this paper, we prove that, with at most O(N ^5/6+ε ) exceptions, all positive integers n ⩽ N can be written as sums of a cube and four cubes of primes.     

A uniqueness result for the motion of micropolar solid–fluid mixtures in unbounded domain

This paper deals with the Eringen’s theory for binary mixtures between elastic micropolar solids and incompressible micropolar fluids (Eringen in J Appl Phys 94:4184–4190, 2003). Using the weighted energy method, an uniqueness result in the case of unbounded domains for small displacement of the solid and for non-slow flow of fluid is presented.     

Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

We consider the time-dependent Landau–Lifshitz–Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii–Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weak–strong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.     

Quantitative Néron theory for torsion bundles

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective curve C _K over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme of r-torsion line bundles on C _K . We determine when there exists a finite R-group scheme, which is a model of over R; in other words, we establish when the Néron model of is finite. The obvious idea would be to study the points of the Néron model over R, but in general these do not represent r-torsion line bundles on a semistable reduction of C _K . Instead, we recast the notion of models on a stack-theoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of C _K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it. A. Chiodo was financially supported by the Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme, MEIF-CT-2003-501940.     

A r -Weighted Poincaré-Type Inequalities for Differential Forms in Some Domains

We prove local weighted integral inequalities for differential forms. Then byusing the local results, we prove global weighted integral inequalities for differential forms in L _s (μ)-averaging domains and in John domains, respectively, which can be considered as generalizations of the classical Poincaré-type inequality.     

A multiplicity result via Ljusternick-Schnirelmann category and Morse theory for a fractional Schrödinger equation in $${\mathbb R^{N}}$$

In this work we study the following class of problems in \({\mathbb R^{N}, N > 2s}\)

$$\varepsilon^{2s}(-\Delta)^{s}u + V(z)u = f(u), \,\,\,u(z) > 0$$

where \({0 < s < 1}\), \({(-\Delta)^{s}}\) is the fractional Laplacian, \({\varepsilon}\) is a positive parameter, the potential \({V : \mathbb{R}^N \to \mathbb{R}}\) and the nonlinearity \({f : \mathbb R \to \mathbb R}\) satisfy suitable assumptions; in particular it is assumed that \({V}\) achieves its positive minimum on some set \({M.}\) By using variational methods we prove existence and multiplicity of positive solutions when \({\varepsilon \to 0^{+}}\). In particular the multiplicity result is obtained by means of the Ljusternick-Schnirelmann and Morse theory, by exploiting the “topological complexity” of the set \({M}\).

     

A “Birkhoff-Lewis” type result for a class of Hamiltonian systems

This paper contains results concerning the existence of long periodic solutions of nonautonomous Hamiltonian systems near the origin.Sponsored by M.P.I. (fondi 60% Problemi diff. nonlineari e teoria dei punti critici fondi 40% Eq. diff. e calcolo delle variazioni)     

A Radial Basis Function Scheme for Option Pricing in Exponential Lévy Models

Abstract

We use Radial Basis Function (RBF) interpolation to price options in exponential Lévy models by numerically solving the fundamental pricing PIDE (Partial integro-differential equations). Our RBF scheme can handle arbitrary singularities of the Lévy measure in 0 without introducing further approximations, making it simpler to implement than competing methods. In numerical experiments using processes from the CGMY-KoBoL class, the scheme is found to be second order convergent in the number of interpolation points, including for processes of unbounded variation.     

A multiplicity result for a generalized Sturm–Liouville problem on time scales

By applying the fixed point theorem in cones, a new and general result on the existence of positive solutions to second-order generalized Sturm–Liouville boundary value problem on a time scale 𝕋 is obtained. The first order Δ-derivative is involved in the nonlinear term f explicitly.