A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

Numerical study of turbulent diesel flow in a pipe with sudden expansion

Three two-equation models and a second-moment closure are implemented in the case of turbulent diesel flow in a pipe with sudden expansion. The chosen two-equation closures are: the standard k–ε, the RNG k–ε and the two-scale k–ε models. The performance of the models is investigated with regard to the non-equilibrium parameter η and the mean strain of the flow, S. Velocity and turbulence kinetic energy predictions of the different models are compared among themselves and with experimental data and are interpreted on the basis of the aforementioned quantities. The effect of more accurate near-wall modeling to the two-equation models is also investigated. The results of the study demonstrate the superiority of the second-moment closure in predicting the flow characteristics over the entire domain. From the two-equation models the RNG derived k–ε model also gave very good predictions, especially when non-equilibrium wall-functions were implemented. As far as η and S are concerned, only the closures with greater physical consistency, such as the two-scale k–ε model, give satisfactory results.     

Homogenization of a nonstationary model equation of electrodynamics

In L ₂(?₃;?₃), we consider a self-adjoint operator ? _ε , ε > 0, generated by the differential expression curl η(x/ε)^?1 curl??ν(x/ε) div. Here the matrix function η(x) with real entries and the real function ν(x) are periodic with respect to some lattice, are positive definite, and are bounded. We study the behavior of the operators cos(τ? _ε ^1/2 ) and ? _ε ^?1/2 sin(τ? _ε ^1/2 ) for τ ∈ ? and small ε. It is shown that these operators converge to cos(τ(?⁰)^1/2) and (?₀)^?1/2 sin(τ(?₀)^1/2), respectively, in the norm of the operators acting from the Sobolev space H ^s (with a suitable s) to ?₂. Here ?⁰ is an effective operator with constant coefficients. Error estimates are obtained and the sharpness of the result with respect to the type of operator norm is studied. The results are used for homogenizing the Cauchy problem for the model hyperbolic equation ? _τ ² v _ε = ?? _ε v _ε , div v _ε = 0, appearing in electrodynamics. We study the application to a nonstationary Maxwell system for the case in which the magnetic permeability is equal to 1 and the dielectric permittivity is given by the matrix η(x/ε).     

An extension of a Bourgain-Lindenstrauss-Milman inequality

Let ‖⋅‖ be a norm on Rn. Averaging ‖(ε₁x₁,…,εnxn)‖ over all the n² choices of , we obtain an expression |||x||| which is an unconditional norm on Rn. Bourgain, Lindenstrauss and Milman [J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis (1986/1987), Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44-66] showed that, for a certain (large) constant η>1, one may average over ηn (random) choices of and obtain a norm that is isomorphic to |||⋅|||. We show that this is the case for any η>1.     

Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary

We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter ε goes to zero. We prove that the family of attractors is upper continuous at the ε=0.     

Asymptotic estimates for the principal eigenvalue of the laplacian in a geodesic ball

LetM be a compact Riemannian manifold and letB _ε be a geodesic ball of radiusε with center0 ∈ M. We investigate the asymptotic behavior ofλ _ε , the principal eigenvalue of the Laplace-Beltrami operator on \(M/\bar B_\varepsilon\) with homogeneous Dirichlet boundary conditions. We prove thatλ _ε ~Cφ _n (ε) wheren = dimM, φ ₂ (ε)=(logε ^?1)^?1 andφ _n (ε) = (n-2)ε ^n-2 (n>2). In the case whereM is a model the constantC is explicitly evaluated.     

Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities

In this paper, the asymptotic behavior of solutions u _ε of the Poisson equation in the ε-periodically perforated domain Ω_ε ? $ {{\mathbb{R}}^n} $ , n ≥ 3, with the third nonlinear boundary condition of the form ? _ν u _ε + ε^?γσ(x, u _ε) = ε ^?γ g(x) on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order ε^α with α > 1 and any γ. Here, all types of asymptotic behavior of solutions u _ε , corresponding to different relations between parameters α and γ, are studied.     

Harmonic approximation of difference operators

For a general class of difference operators Hε=Tε+Vε on ?²(d(εZ)), where Vε is a multi-well potential and ε is a small parameter, we analyze the asymptotic behavior as ε→0 of the (low-lying) eigenvalues and eigenfunctions. We show that the first n eigenvalues of Hε converge to the first n eigenvalues of the direct sum of harmonic oscillators on Rd located at the several wells. Our proof is microlocal.     

Derivation of a quasi‐stationary coupled Darcy–Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width η_ε. Passing to the limit when ε goes to zero, we find a critical size in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

Homogenization of variational inequalities for a biharmonic operator with constraints on subsets <Emphasis Type="Italic">ε</Emphasis>-periodically placed along a manifold of large dimension

Behavior of solutions of variational inequalities for a biharmonic operator is studied. These inequalities correspond to one-sided constraints on subsets of a domain Ω placed ε-periodically. All possible behavior types of solutions u _ε of variational inequalities are considered for ε → 0 depending on relations between small parameters, which are the structure period ε and the contraction coefficient a _ε of subsets where one-sided constraints are posed.     

The properties of a family of tiles with a parameter

This paper studies topological and tiling properties of a family of self-affine fractal tiles with a real parameter ε. Both interior and boundary structures have been discussed. Moreover, connectedness and tiling properties of the tile Tε are proved to be dependent upon ε.     

On a hierarchy of models for electrical conduction in biological tissues

In this paper we derive a hierarchy of models for electrical conduction in a biological tissue, which is represented by a periodic array of period ε of conducting phases surrounded by dielectric shells of thickness εη included in a conductive matrix. Such a hierarchy will be obtained from the Maxwell equations by means of a concentration process η → 0, followed by a homogenization limit with respect to ε. These models are then compared with regard to their physical meaning and mathematical issues. Copyright © 2005 John Wiley & Sons, Ltd.     

Homogenization of the equations of state for a heterogeneous layered medium consisting of two creep materials

A mathematical model that describes the joint motion of periodically alternating layers of two isotropic creep materials is considered. It is assumed that all layers are parallel to one of the coordinate planes and the thickness of any two adjacent layers is ε. For this model, the corresponding homogenized model for ε → 0 is constructed, which describes the behavior of a homogeneous creep material.     

Étude asymptotique d'un modèle simple de flammes prémélangées dans un domaine extérieur de RN

We consider a semilinear elliptic equation ?ΔT_ε+u·?T_ε=f_ε(T_ε)(1?T_ε) in outer domains of $\text{R}{}^{\mathrm{N}}$ with Dirichlet's boundary conditions. This Note deals with the questions of existence, uniqueness and the asymptotic behavior of solutions T_ε as ε tends to 0 and the reaction term behaves as a Dirac distribution. Such problems arise in the modelling of premixed flames in the limit of high activations energies. To cite this article: G. Sagon, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On needed reals

Given a binary relationR, we call a subsetA of the range ofR R-adequate if for everyx in the domain there is someyεA such that (x, y)εR. Following Blass [4], we call a realη ”needed” forR if in everyR-adequate set we find an element from whichη is Turing computable. We show that every real needed for inclusion on the Lebesgue null sets,Cof(\(\mathcal{N}\)), is hyperarithmetic. Replacing “R-adequate” by “R-adequate with minimal cardinality” we get the related notion of being “weakly needed”. We show that it is consistent that the two notions do not coincide for the reaping relation. (They coincide in many models.) We show that not all hyperarithmetic reals are needed for the reaping relation. This answers some questions asked by Blass at the Oberwolfach conference in December 1999 and in [4].     

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ³ is considered. It is assumed that the permittivity η^ε(x)=η(ε⁻x), ε > 0, is a rapidly oscillating positive matrix function and the permeability μ₀ is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L ₂(ℝ³)-norm with order-sharp remainder estimates. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 3–23, 2007 Original Russian Text Copyright ? by M. Sh. Birman and T. A. Suslina Dedicated to the memory of the great mathematician Mark Grigor’evich Krein Supported by RFBR grants no. 05-01-01076-a, 05-01-02944-YaF-a.     

On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong-Loewy conjecture

Let C={1,2,…,m} and f be a multiplicative function such that (f∗μ)(k)>0 for every positive integer k and the Euler product converges. Let (C_f)=(f(i,j)) be the m×m matrix defined on the set C having f evaluated at the greatest common divisor (i,j) of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of (C_f)^-1, the inverse of (C_f), in terms of ζ_f. Specializing these bounds for the arithmetical functions f=N^ε,J_ε and σ_ε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (C_N^ε),(C_Jε) and (C_σε) depending on ε when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy [S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551-569].     

Layered stable equilibria of a reaction-diffusion equation with nonlinear Neumann boundary condition

In this work we investigate the existence and asymptotic profile of a family of layered stable stationary solutions to the scalar equation ut=ε²Δu+f(u) in a smooth bounded domain Ω⊂R³ under the boundary condition εν_∂u=δεg(u). It is assumed that Ω has a cross-section which locally minimizes area and limε→0εlnδε=κ, with 0?κ<∞ and δε>1 when κ=0. The functions f and g are of bistable type and do not necessarily have the same zeros what makes the asymptotic geometric profile of the solutions on the boundary to be different from the one in the interior.     

Analysis of a finite-difference scheme for a singularly perturbed problem with two small parameters

We study a model linear convection-diffusion-reaction problem where both the diffusion term and the convection term are multiplied by small parameters ε_d and ε_c, respectively. Depending on the size of the parameters the solution of the problem may exhibit exponential layers at both end points of the domain. Sharp bounds for the derivatives of the solution are derived using a barrier-function technique. These bounds are applied in the analysis of a simple upwind-difference scheme on Shishkin meshes. This method is established to be almost first-order convergent, independently of the parameters ε_d and ε_c.