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The Pauli-Poisson equation is a semi-relativistic model for charged spin-1∕2-par-ticles in a strong external magnetic field and a self-consistent electric potential computed from the Poisson equation in three space dimensions. It is a system of two magnetic Schrödinger equations for the two components of the Pauli 2-spinor, representing the two spin states of a fermion, coupled by the additional Stern-Gerlach term representing the interaction of magnetic field and spin. We study the global well-posedness in the energy space and the semiclassical limit of the Pauli-Poisson to the magnetic Vlasov-Poisson equation with Lorentz force and the semiclassical limit of the linear Pauli equation to the magnetic Vlasov equation with Lorentz force. We use Wigner transforms and a density matrix formulation for mixed states, extending the work of P. L. Lions & T. Paul as well as P. Markowich & N.J. Mauser on the semiclassical limit of the non-relativistic Schrödinger-Poisson equation. [ABSTRACT FROM AUTHOR]

In this article, we investigate the behavior of weak solutions for the three-dimensional Navier-Stokes-Voigt-Brinkman-Forchheimer fluid model with memory and Tresca friction law within a thin domain. We analyze the asymptotic behavior as one dimension of the fluid domain approaches zero. We derive the limit problem and obtain the specific Reynolds equation, while also establishing the uniqueness of the limit velocity and pressure distributions.

We look at computational physics from an electrical engineering perspective and suggest that several concepts of mathematics, not so well-established in computational physics literature, present themselves as opportunities in the field. We discuss elliptic complexes and highlight the category theoretical background and its role as a unifying language between algebraic topology, differential geometry, and modelling software design. In particular, the ubiquitous concept of naturality is central. Natural differential operators have functorial analogues on the cochains of triangulated manifolds. In order to establish this correspondence, we derive formulas involving simplices and barycentric coordinates, defining discrete vector fields and a discrete Lie derivative as a result of a discrete analogue of Cartan’s magic formula. This theorem is the main mathematical result of the paper. [ABSTRACT FROM AUTHOR]

In this paper, we deal with an inverse boundary value problem for the Maxwell equations with boundary data assumed known only in accessible part Γ of the boundary. We aim to prove uniqueness results using the Dirichlet to Neumann data with measurements limited to an open part of the boundary and we seek to reconstruct the complex refractive index n in the interior of a body. Further, using the impedance map restricted to Γ , we may identify locations of small volume fraction perturbations of the refractive index. [ABSTRACT FROM AUTHOR]

The aim of this work is the study of a nonlinear boundary value problem which theoretically generalizes the Lamé system with disturbance in a thin 3D domain with friction and a generalized boundary condition. For the resolution of the considered problem and after the variational formulation, we construct an operator from the variational problem. Then we prove that this operator has certain properties which allows us to apply the theorem of existence and uniqueness of the solution of variational inequalities of the 2nd kind. Finally, using a change of scale, we transport the variational problem to an equivalent problem defined on a domain independent of the parameter ζ {{\zeta}} and subsequently we obtain the limit problem and the generalized weak equations of the initial problem. [ABSTRACT FROM AUTHOR]

Mathematical models of polyelectrolyte gels are often simplified by assuming the gel is electrically neutral. The rationale behind this assumption is that the thickness of the electric double layer (EDL) at the free surface of the gel is small compared to the size of the gel. Hence, the thin-EDL limit is taken, in which the thickness of the EDL is set to zero. Despite the widespread use of the thin-EDL limit, the solutions in the EDL are rarely computed and shown to match to the solutions for the electrically neutral bulk. The aims of this paper are to study the structure of the EDL and establish the validity of the thin-EDL limit. The model for the gel accounts for phase separation, which gives rise to diffuse interfaces with a thickness described by the Kuhn length. We show that the solutions in the EDL can only be asymptotically matched to the solutions for an electrically neutral bulk, in general, when the Debye length is much smaller than the Kuhn length. If the Debye length is similar to or larger than the Kuhn length, then phase separation can be initiated in the EDL. This phase separation spreads into the bulk of the gel and gives rise to electrically charged layers with different degrees of swelling. Thus, the thin-EDL limit and the assumption of electroneutrality only generally apply when the Debye length is much smaller than the Kuhn length. [ABSTRACT FROM AUTHOR]

Three-dimensional boundary-value problem describing a generalized non-Newtonian fluid with nonlinear Tresca friction type in a thin domain Ω ε are considered. We study the asymptotic behavior when one dimension of the fluid domain tends to zero. We prove some weak convergence of the velocity and the pressure of the fluid. Then the limit problem in two-dimensional domain and the specific Reynolds equation are obtained. [ABSTRACT FROM AUTHOR]

This article establishes an asymptotic behavior for the elasticity systems with nonlinear source and dissipative terms in a three-dimensional thin domain, which generalizes some previous works. We consider the limit when the thickness tends to zero, and we prove that the limit solution u∗{u}^{\ast } is a solution of a two-dimensional boundary value problem with lower Tresca’s free-boundary conditions. Moreover, we obtain the weak Reynolds-type equation.

In this work, we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon a hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objectives is to validate numerically generalized telegrapher's models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article. [ABSTRACT FROM AUTHOR]

In this paper, we provide further spectral analysis of the general asymptotic scattering resonances formula of small 3D dielectrics of arbitrary shape with high contrast, already derived in other works to a first-order approximation. To investigate the components of a full expansion of such resonances, a breakdown is presented for the case of high-contrast nanospheres, in terms of its radius h in the interval [0, 1]. We also derive, for radially symmetric fields, an exact resonance formula for a spherical scatterer in terms of its radius, not necessarily small, and dielectric susceptibility coefficient, not necessarily high. This formula is further developed and simplified in the case of high contrast nanospheres. Such formulas are useful in imaging applications to identify objects’ properties from frequency measurements. Another application is the study of negative refractive index materials, such as metamaterials, and the anomalous localized resonance phenomenon (ALR) that is associated with cloaking and superlensing. [ABSTRACT FROM AUTHOR]

The high order homogenization techniques potentially generate the so-called infinite order homogenized equations. Since long ago, the coefficients at higher order derivatives in these equations have been calculated within various refined theories for both periodic composites and thin structures. However, it was not always clear, what is a well-posed mathematical formulation for such equations. In the present paper, we discuss two techniques for constructing a second-order homogenized equation. One of them is concerned with the projection of a weak formulation of the original problem on an 'ansatz subspace'. The second one corresponds to the traditional two scale asymptotic expansion using the representation of a second-order corrector via the solution of the classical (leading order) homogenized equation. [ABSTRACT FROM AUTHOR]

We consider the propagation of acoustic waves in a 2D waveguide unbounded in one direction and containing a compact obstacle. The wavenumber is fixed so that only one mode can propagate. The goal of this work is to propose a method to cloak the obstacle. More precisely, we add to the geometry thin outer resonators of width ε and we explain how to choose their positions as well as their lengths to get a transmission coefficient approximately equal to one as if there were no obstacle. In the process, we also investigate several related problems. In particular, we explain how to get zero transmission and how to design phase shifters. The approach is based on asymptotic analysis in presence of thin resonators. An essential point is that we work around resonance lengths of the resonators. This allows us to obtain effects of order one with geometrical perturbations of width ε . Various numerical experiments illustrate the theory. [ABSTRACT FROM AUTHOR]

Interchanging the role of space and time is widely used in nonlinear optics for modeling the evolution of light pulses in glass fibers. A phenomenological model for the mathematical description of light pulses in glass fibers with a periodic structure in this set-up is the so-called dispersion management equation. It is the purpose of this paper to answer the question whether the dispersion management equation or other modulation equations are more than phenomenological models in this situation. Using Floquet theory we prove that in case of comparable wave lengths of the light and of the fiber periodicity the NLS equation and NLS like modulation equations with constant coefficients can be derived and justified through error estimates under the assumption that rather strong stability and non-resonance conditions hold. This is the first NLS approximation result documented for time-periodic dispersive systems. We explain that the failure of these conditions allows us to prove that these modulation equations in general make wrong predictions. The reasons for this failure and the behavior of the system for a fiber periodicity much larger than the wave length of light shows that interchanging the role of space and time for glass fibers with a periodic structure leads to unwanted phenomena. [ABSTRACT FROM AUTHOR]

The purpose of this paper is the study of the asymptotic modeling of a linearly thin elastic isotropic membrane shell which is subjected to frictionless contact with a rigid obstacle on part of the boundary. Using the formal asymptotic expansions method, we obtain a two dimensional Signorini membrane shell model, then we justify the result by convergence theorem. [ABSTRACT FROM AUTHOR]

Understanding the dynamics underlying fluid transport in tumour tissues is of fundamental importance to assess processes of drug delivery. Here, we analyse the impact of the tumour microscopic properties on the macroscopic dynamics of vascular and interstitial fluid flow. More precisely, we investigate the impact of the capillary wall permeability and the hydraulic conductivity of the interstitium on the macroscopic model arising from formal asymptotic 2-scale techniques. The homogenization technique allows us to derive two macroscale tissue models of fluid flow that take into account the microscopic structure of the vessels and the interstitial tissue. Different regimes were derived according to the magnitude of the vessel wall permeability and the interstitial hydraulic conductivity. Importantly, we provide an analysis of the properties of the models and show the link between them. Numerical simulations were eventually performed to test the models and to investigate the impact of the microstructure on the fluid transport. Future applications of our models include their calibration with real imaging data to investigate the impact of the tumour microenvironment on drug delivery. [ABSTRACT FROM AUTHOR]

This paper is concerned with the design of two different classes of Galerkin boundary element methods for the solution of high-frequency sound-hard scattering problems in the exterior of two-dimensional smooth convex scatterers. We prove in this paper that both methods require a small increase (in the order of k ϵ for any ϵ > 0 ) in the number of degrees of freedom to guarantee frequency independent precisions with increasing wavenumber k. In addition, the accuracy of the numerical solutions are independent of frequency provided sufficiently many terms in the asymptotic expansion are incorporated into the integral equation formulation. Numerical results validating O (k ϵ) algorithms are presented. [ABSTRACT FROM AUTHOR]

The quasistatic problem of a viscoelastic body in a three-dimensional thin domain with Tresca's friction law is considered. The viscoelasticity coefficients and data for this system are assumed to vary with respect to the thickness ε. The asymptotic behavior of weak solution, when ε tends to zero, is proved, and the limit solution is identified in a new data system. We show that when the thin layer disappears, its traces form a new contact law between the rigid plane and the viscoelastic body. In which case, a generalized weak form equation is formulated, the uniqueness result for the limit problem is also proved. [ABSTRACT FROM AUTHOR]

This paper gives a brief overview of some configurations in which high-frequency wave propagation modelled by Helmholtz equation gives rise to solutions that vary rapidly across thin layers. The configurations are grouped according to their mathematical structure and tractability and one of them concerns a famous open problem of mathematical physics. [ABSTRACT FROM AUTHOR]

We consider an array of double oligomers in an optical waveguide device. A mathematical model for the system is the coupled discrete nonlinear Schrödinger equations, where the gain-and-loss parameter contributes to the complex-valued linear coupling. The array caters to an optical simulation of the parity-time (PT )-symmetry property between the coupled arms. The system admits fundamental bright discrete soliton solutions. We investigate their existence and spectral stability using perturbation theory analysis. These analytical findings are verified further numerically using the Newton–Raphson method and a standard eigenvalue-problem solver. Our study focuses on two natural discrete modes of the solitons: single- and double-excited-sites, also known as onsite and intersite modes, respectively. Each of these modes acquires three distinct configurations between the dimer arms, i.e., symmetric, asymmetric, and antisymmetric. Although both intersite and onsite discrete solitons are generally unstable, the latter can be stable, depending on the combined values of the propagation constant, horizontal linear coupling coefficient, and gain–loss parameter. [ABSTRACT FROM AUTHOR]

Excitation of surface-plasmon resonances of closely spaced nanometallic structures is a key technique used in nanoplasmonics to control light on subwavelength scales and generate highly confined electric-field hotspots. In this paper, we develop asymptotic approximations in the near-contact limit for the entire set of surface-plasmon modes associated with the prototypical sphere dimer geometry. Starting from the quasi-static plasmonic eigenvalue problem, we employ the method of matched asymptotic expansions between a gap region, where the boundaries are approximately paraboloidal, pole regions within the spheres and close to the gap, and a particle-scale region where the spheres appear to touch at leading order. For those modes that are strongly localised to the gap, relating the gap and pole regions gives a set of effective eigenvalue problems formulated over a half space representing one of the poles. We solve these problems using integral transforms, finding asymptotic approximations, singular in the dimensionless gap width, for the eigenvalues and eigenfunctions. In the special case of modes that are both axisymmetric and odd about the plane bisecting the gap, where matching with the outer region introduces a logarithmic dependence upon the dimensionless gap width, our analysis follows Schnitzer [Singular perturbations approach to localized surface-plasmon resonance: nearly touching metal nanospheres. Phys. Rev. B92(23), 235428 (2015)]. We also analyse the so-called anomalous family of even modes, characterised by field distributions excluded from the gap. We demonstrate excellent agreement between our asymptotic formulae and exact calculations. [ABSTRACT FROM AUTHOR]

We consider high-frequency multiple-scattering problems in the exterior of two-dimensional smooth scatterers consisting of finitely many compact, disjoint, and strictly convex obstacles. To deal with this problem, we propose Galerkin boundary element methods, namely the frequency-adapted Galerkin boundary element methods and Galerkin boundary element methods generated using frequency-dependent changes of variables. For both of these new algorithms, in connection with each multiple-scattering iterate, we show that the number of degrees of freedom needs to increase as O (k ϵ) (for any ϵ > 0 ) with increasing wavenumber k to attain frequency-independent error tolerances. We support our theoretical developments by a variety of numerical implementations. [ABSTRACT FROM AUTHOR]

For the case of singularly perturbed elliptic transmission problems we demonstrate the use of asymptotic expansion techniques both for establishing regularity results for the solution and for deriving a priori error estimates for boundary element Galerkin discretisation. The dependence of the corresponding bounds on the singular perturbation parameter is studied in detail. This dependence clearly manifests itself in numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we study the asymptotic behavior of the solutions of time dependent micromagnetism problem in a multi-domain consisting of a thin-wire in junction with a thin film. We assume that the volumes of the two parts composing each multi-structure vanish with same rate. We obtain a 1 D limit problem on the thin-wire and a 2 D limit problem on the thin film, and the two limit problems are uncoupled. The limit problem remains non-convex, but now it becomes completely local. [ABSTRACT FROM AUTHOR]

In this paper, we consider a non-linear problem in a stationary regime in a three-dimensional thin domain $\Omega^{\varepsilon}$ with Fourier and Tresca boundary conditions. In the first step, we derive a variational formulation of the mechanical problem. We then study the asymptotic behavior in the one dimension case when the domain parameter tends to zero. In the latter case, a specific Reynolds equation associated with variational inequalities is obtained and the uniqueness of the limit velocity and pressure are proved. [ABSTRACT FROM AUTHOR]

Asymptotic expansions of the voltage potential in terms of the 'radius' of a diametrically small (or several diametrically small) material inhomogeneity(ies) are by now quite well-known. Such asymptotic expansions for diametrically small inhomogeneities are uniform with respect to the conductivity of the inhomogeneities. In contrast, thin inhomogeneities, whose limit set is a smooth, codimension 1 manifold, σ, are examples of inhomogeneities for which the convergence to the background potential, or the standard expansion cannot be valid uniformly with respect to the conductivity, a, of the inhomogeneity. Indeed, by taking a close to 0 or to infinity, one obtains either a nearly homogeneous Neumann condition or nearly constant Dirichlet condition at the boundary of the inhomogeneity, and this difference in boundary condition is retained in the limit. The purpose of this paper is to find a 'simple' replacement for the background potential, with the following properties: (1) This replacement may be (simply) calculated from the limiting domain Ωσ, the boundary data on the boundary of Ω, and the right-hand side. (2) This replacement depends on the thickness of the inhomogeneity and the conductivity, a, through its boundary conditions on σ. (3) The difference between this replacement and the true voltage potential converges to 0 uniformly in a, as the inhomogeneity thickness tends to 0. [ABSTRACT FROM AUTHOR]

The asymptotic approximation of the displacement of a linear elastic beam with prescribed non-homogeneous Dirichlet conditions at an end is constructed. The non-homogeneous Dirichlet conditions considered in this work have the form of rigid displacements. The six values needed to state an auxiliary one-dimensional system are obtained and the error of the approximation is estimated. A numerical example illustrates the high precision of the approximation. [ABSTRACT FROM PUBLISHER]

Cloaking using complementary media was suggested by Lai et al. in [8] . The study of this problem faces two difficulties. Firstly, this problem is unstable since the equations describing the phenomenon have sign changing coefficients, hence the ellipticity is lost. Secondly, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this paper, we give a proof of cloaking using complementary media for a class of schemes inspired from [8] in the quasistatic regime. To handle the localized resonance, we introduce the technique of removing localized singularity and apply a three spheres inequality. The proof also uses the reflecting technique in [11] . To our knowledge, this work presents the first proof on cloaking using complementary media. [ABSTRACT FROM AUTHOR]

The aim of the work described in this paper is to determine, via an asymptotic analysis, the limiting form of the free energy governing in the first case 3D ferromagnetic nanowires of infinite length in the limit and in the second case 3D thin films which become infinite when their thickness is vanished. A 1D limit problem on the nanowires and a 2D limit problem on the thin films are obtained. [ABSTRACT FROM AUTHOR]

Starting from a $$nD, n\ge 2$$ , non-convex and nonlocal micromagnetic energy, we determine, via an asymptotic analysis, the free energy of a $$pD$$ ferromagnetic domain, $$1\le p

In this paper we propose and analyse a hybrid numerical-asymptotic boundary element method for the solution of problems of high frequency acoustic scattering by a class of sound-soft nonconvex polygons. The approximation space is enriched with carefully chosen oscillatory basis functions; these are selected via a study of the high frequency asymptotic behaviour of the solution. We demonstrate via a rigorous error analysis, supported by numerical examples, that to achieve any desired accuracy it is sufficient for the number of degrees of freedom to grow only in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods. This appears to be the first such numerical analysis result for any problem of scattering by a nonconvex obstacle. Our analysis is based on new frequency-explicit bounds on the normal derivative of the solution on the boundary and on its analytic continuation into the complex plane. [ABSTRACT FROM AUTHOR]

This work consists in the asymptotic analysis of the solution of Poisson equation in a bounded domain of RP(P= 2, 3) with a thin layer. We use a method based on hierarchical variational equations to derive an explicitly asymptotic expansion of the solution with respect to the thickness of the thin layer. We determine the first two terms of the expansion and prove the error estimate made by truncating the expansion after a finite number of terms. Next, using the first two terms of the asymptotic expansion, we show that we can model the effect of the thin layer by a problem with transmission conditions of order two. [ABSTRACT FROM PUBLISHER]