bluuebunny/crossref_metadata_2025 huggingface dataset:
Klein paradox and scattering theory for the semi-classical Dirac equation
Authors: Khochman Abdallah
Published: November 2019

We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.
Topological sensitivity analysis with respect to a small hole located at the boundary of the domain
Authors: Samet Bessem
Published: November 2019

The topological sensitivity analysis consists to study the variation of a shape functional with respect to emerging of small holes in the interior of the domain occupied by the body. In this paper, we consider the case of a hole located at the boundary of the domain. The topological asymptotic expansion is derived for the Laplace equation in three cases: Dirichlet condition, Neumann condition and Robin condition on the boundary of the hole.
On approximate solutions to the wavefront speed of reaction–diffusion–convection problem in nonuniform media
Authors: Rocha E.M. , Rodrigues M.M.
Published: November 2019

A class of perturbed reaction–diffusion–convection (R–D–C) equations with reaction, diffusion and convection functions in nonuniform media is studied. By using singular perturbation analysis and Hamilton–Jacobi theory, the wavefront speed of such R–D–C equations is computed for different classes of nonlinearities.
Soliton dynamics for CNLS systems with potentials
Authors: Montefusco Eugenio , Pellacci Benedetta , Squassina Marco
Published: November 2019

The semiclassical limit of a weakly coupled nonlinear focusing Schrödinger system in presence of a nonconstant potential is studied. The initial data is of the form (u1, u2) with ui=ri ((x−x˜)/ε)e(i/ε)x·ξ˜, where (r1, r2) is a real ground state solution, belonging to a suitable class, of an associated autonomous elliptic system. For ε sufficiently small, the solution (ϕ1, ϕ2) will been shown to have, locally in time, the form (r1 ((x−x(t))/ε)e(i/ε)x·ξ(t), r2 ((x−x(t))/ε)e(i/ε)x·ξ(t)), where (x(t), ξ(t)) is the solution of the Hamiltonian system x˙(t)=ξ(t), ξ˙(t)=−∇V(x(t)) with x(0)=x˜ and ξ(0)=ξ˜.
Nonautonomous initial-boundary value problems in the viscoplasticity theory
Authors: Pompe Waldemar
Published: November 2019

We prove existence of a solution to the nonautonomous, noncoercive, quasistatic models in the viscoplasticity theory. This extends the results shown earlier by the author in Math. Methods Appl. Sci. 27 (2004), 1347–1365 and Math. Methods Appl. Sci. 31 (2008), 775–792, for the autonomous case. As a special case we obtain existence of the solution to the nonautonomous kinematic hardening model, where the constitutive nonlinearity has a polynomial or a rapid growth at infinity.
Asymptotics for a generalization of Hermite polynomials
Authors: Alfaro Manuel , Peña Ana , Rezola M. Luisa , Moreno-Balcázar Juan José
Published: November 2019

We consider a generalization of the classical Hermite polynomials by the addition of terms involving derivatives in the inner product. This type of generalization has been studied in the literature from the point of view of the algebraic properties. Thus, our aim is to study the asymptotics of this sequence of nonstandard orthogonal polynomials. In fact, we obtain Mehler–Heine type formulas for these polynomials and, as a consequence, we prove that there exists an acceleration of the convergence of the smallest positive zeros of these generalized Hermite polynomials towards the origin.
Criteria for Bochner's extension problem
Authors: Ruzhansky Michael , Sugimoto Mitsuru
Published: November 2019

A necessary and sufficient condition for the resolution of the weak extension problem is given. This criterion is applied to also give a criterion for the solvability of the classical Bochner's extension problem in the Lp-category. The solution of the Lp-extension problem by Bochner [J. Math. Pures Appl. 35 (1956), 193–202] giving the relation between the order of the operator, the dimension, and index p, for which the Lp-extension property holds, can be viewed as a subcritical case of the general Lp-extension problem. In general, this property fails in some critical and in all supercritical cases. In this paper, the Lp-extension problem is investigated for operators of all orders and for all 1≤p≤∞. Necessary and sufficient conditions on the subset of Lp are given for which the Lp-extension property still holds, in the critical and supercritical cases.
A spectral study of the boundary controllability of the linear 2-D wave equation in a rectangle
Authors: Micu Sorin , de Teresa Luz
Published: November 2019

The paper studies the controllability properties of the linear 2-D wave equation in the rectangle Ω=(0, a)×(0, b). We consider two types of action, on an edge or on two adjacent edges of the boundary. Our analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency. In both analyzed cases we give a Fourier characterization of the controllable spaces of initial data and we construct particular controls for them.
Amplitude equations and asymptotic expansions for multi-scale problems
Authors: Kirkinis Eleftherios
Published: November 2019

In this paper we introduce a new technique to obtain the slow-motion dynamics in nonequilibrium and singularly perturbed problems characterized by multiple scales. Our method is based on a straightforward asymptotic reduction of the order of the governing differential equation and leads to amplitude equations that describe the slowly-varying envelope variation of a uniformly valid asymptotic expansion. This may constitute a simpler and in certain cases a more general approach toward the derivation of asymptotic expansions, compared to other mainstream methods such as the method of Multiple Scales or Matched Asymptotic expansions because of its relation with the Renormalization Group. We illustrate our method with a number of singularly perturbed problems for ordinary and partial differential equations and recover certain results from the literature as special cases.
Large time behavior of solutions of viscous Hamilton–Jacobi equations with superquadratic Hamiltonian
Authors: Tabet Tchamba Thierry
Published: November 2019

We study the long-time behavior of the unique viscosity solution u of the viscous Hamilton–Jacobi equation ut−Δu+|Du|m=f in Ω×(0, +∞) with inhomogeneous Dirichlet boundary conditions, where Ω is a bounded domain of RN. We mainly focus on the superquadratic case (m>2) and consider the Dirichlet conditions in the generalized viscosity sense. Under rather natural assumptions on f, the initial and boundary data, we connect the problem studied to its associated stationary generalized Dirichlet problem on one hand and to a stationary problem with a state constraint boundary condition on the other hand.
Zigzag nanoribbons in external electric fields
Authors: Korotyaev Evgeny L. , Kutsenko Anton
Published: November 2019

We consider the Schrödinger operator on nanoribbons (tight-binding models) in an external electric potential V on the plane. The corresponding electric field is perpendicular to the axis of the nanoribbon. If V=0, then the spectrum of the Schrödinger operator consists of two non-flat bands and one flat band (an eigenvalue with infinite multiplicity) between them. If we switch on a weak electric potential V→0, then there are two cases: (1) this eigenvalue splits into the small spectral band. We determine the asymptotics of the spectral bands for small fields. (2) the unperturbed eigenvalue remains the flat band. We describe all potentials when the unperturbed eigenvalue remains the flat band and when one splits into the small band of the continuous spectrum.
Estimates in homogenization of parabolic equations with locally periodic coefficients
Authors: Pastukhova Svetlana
Published: November 2019

We study parabolic equations with highly inhomogeneous locally periodic coefficients. The small parameter ε defines the degree of inhomogenity. The difference between the operator exponentials corresponding to the initial and the homogenized equations is of the main interest. We prove the estimate for this difference in operator L2-norm of order ε1/2.
Non-linear asymptotic stability for the through-passing flows of inviscid incompressible fluid
Authors: Morgulis A.
Published: November 2019

The paper addresses the dynamics of inviscid incompressible fluid confined within bounded domain with the inflow and outflow of fluid through certain parts of the boundary. This system is non-conservative essentially since the fluxes of energy and vorticity through the flow boundary are not equal to zero. Therefore, the dynamics of such flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of instability or the excitation of the self-oscillations, etc. These phenomena are studied extensively for the flows of the viscous fluids but not for the inviscid ones. In this paper, we prove a sufficient condition for the non-linear asymptotic stability of the inviscid steady flows.
Asymptotic analysis and diffusion limit of the Persistent Turning Walker Model
Authors: Cattiaux Patrick , Chafaï Djalil , Motsch Sébastien
Published: November 2019

The Persistent Turning Walker Model (PTWM) was introduced by Gautrais et al. in Mathematical Biology for the modelling of fish motion. It involves a nonlinear pathwise functional of a non-elliptic hypo-elliptic diffusion. This diffusion solves a kinetic Fokker–Planck equation based on an Ornstein–Uhlenbeck Gaussian process. The long time “diffusive” behavior of this model was recently studied by Degond and Motsch using partial differential equations techniques. This model is however intrinsically probabilistic. In the present paper, we show how the long time diffusive behavior of this model can be essentially recovered and extended by using appropriate tools from stochastic analysis. The approach can be adapted to many other kinetic “probabilistic” models.
Homogenizing media containing a highly conductive honeycomb substructure
Authors: Gruais Isabelle , Poliševski Dan
Published: November 2019

The present paper deals with the homogenization of the heat conduction which takes place in a binary three-dimensional medium consisting of an ambiental phase having conductivity of unity order and a rectangular honeycomb structure formed by a set of thin layers crossing orthogonally and periodically. We consider the case when the conductivity of the thin layers is in inverse proportion to the vanishing volume of the rectangular honeycomb structure. We find the system that governs the asymptotic behaviour of the temperature distribution of this binary medium. The dependence with respect to the thicknesses of the layers is also emphasized. We use an energetic method associated to a natural control-zone of the vanishing domain.
Homogenization of an elasticity problem for a catalyst support by using the unfolding method
Authors: Griso G. , Sánchez M.T.
Published: November 2019

The goal of this work is to study the asymptotic behaviour of catalyst supports in a linear elasticity problem. The catalyst support is a structure made of beams, placed periodically and with inner holes. We introduce a decomposition of the displacements in such a structure and we prove some convergence results. Finally, we obtain the limit problem when the elasticity parameter e and the periodicity parameter ε go to zero.
Selective decay principle for 2D magnetohydrodynamic flows
Authors: Zhan Mei-Qin
Published: November 2019

The selective decay phenomena has been observed by physicists for many dynamic flows such as Navier–Stokes flows, barotropic geophysical flows, and magnetohydrodynamic (MHD) flows in either actual physical experiments or numerical simulations. Rigorous mathematical works have been carried out for both Navier–Stokes and barotropic geophysical flows. It is the goal of this paper to verify the selective decay principle mathematically for the 2D magnetohydrodynamic (MHD) flows.
A spectral analogue of the Meinardus theorem on asymptotics of the number of partitions
Authors: Tate Tatsuya
Published: November 2019

We discuss asymptotics of the number of states of Boson gas whose Hamiltonian is given by a positive elliptic pseudo-differential operator of order one on a compact manifold. We obtain an asymptotic formula for the average of the number of states. Furthermore, when the operator has integer eigenvalues and the periodic orbits of period less than 2π of the classical mechanics form clean submanifolds of lower dimensions, we give an asymptotic formula for the number of states itself. This is regarded as an analogue of the Meinardus theorem on asymptotics of the number of partitions of a positive integer. We use the Meinardus saddle point method of obtaining the asymptotics of the number of partitions, combined with a theorem due to Duistermaat–Guillemin and other authors on the singularities of the trace of the wave operators.
Scaling up of an underground nuclear waste repository including a possibly damaged zone
Authors: Bourgeat Alain , Marušić-Paloka Eduard , Piatnitski Andrey
Published: November 2019

We derive a macroscopic model for an underground nuclear waste repository consisting of long storage cells linked by a possibly damaged drifts. As the first result we find a simple first-order approximation. Secondly, we compute a corrector using a matched expansion around the drift. We prove an appropriate convergence result.
On the spectrum of the Neumann problem for Laplace equation in a domain with a narrow slit
Authors: Gadyl'shin Rustem R. , Il'in Arlen M.
Published: November 2019

The Neumann problem in two-dimensional domain with a narrow slit is studied. The width of the slit is a small parameter 0<ε�1. The complete asymptotic expansion with respect to ε for the eigenvalue of the perturbed problem converging to a simple eigenvalue of the limiting problem is constructed by means of the method of the matching asymptotic expansions. It is shown that the regular perturbation theory can formally be applied in a natural way up to terms of order ε2. However, the result obtained in that way is false. The correct result can be obtained only by means of inner asymptotic expansion.

We also show that the eigenvalue of the perturbed problem can be both more and less than the eigenvalue of the limiting problem subject to the position and geometry of the slit.
Periodic constant-sign solutions for systems of Hill's equations
Authors: Agarwal Ravi P. , O'Regan Donal , Wong Patricia J.Y.
Published: November 2019

We consider the system of Hill's equations

u″i(t)+ai(t)ui(t)=Fi(t, u1(t), u2(t), …, un(t)), 1≤i≤n,

where ai and Fi are periodic in t, and the non-linearities Fi(t, x1, x2, …, xn) can be singular at xj=0 where j∈{1, 2, …, n}. Criteria are offered for the existence of periodic constant-sign solutions, i.e., θiui(t)≥0 for each 1≤i≤n, where θi∈{1, −1} is fixed. The main tool used is Schauder's fixed point theorem. We also include examples to illustrate the usefulness of the results obtained.
On the semiclassical 3D Neumann Laplacian with variable magnetic field
Authors: Raymond Nicolas
Published: November 2019

In this paper we are interested in the semiclassical estimates of the spectrum of the Neumann Laplacian in dimension 3. This work aims to present a complementary case to the one presented in the paper of Helffer and Morame in the case of constant magnetic field. More precisely, in the case when the magnetic field is variable and under the most generic condition for which boundary localizations can be observed, we prove a three terms upper bound for the lowest eigenvalue and establish some semiclassical behaviour of the spectrum.
Integral operators induced by Bergman type kernels in the half plane
Authors: Dostanić Milutin R.
Published: November 2019

We determine the norm of a class of integral operators induced by the reproducing kernel of Bergman spaces in upper half plane.
Asymptotic expansion of L2-norms of solutions to the heat and dissipative wave equations
Authors: Volkmer Hans
Published: November 2019

Let v(t, x) and u(t, x) be solutions of the heat equation vt−Δv=0 and dissipative wave equation utt+ut−Δu=0, respectively. The paper finds the asymptotic expansions of the squared L2-norms of v, u and u−v as well as of their derivatives as t→∞. Suitable conditions on the initial values u(0, x), ut(0, x) and v(0, x) lead to cancellation of the leading terms of the asymptotic expansion of u−v explaining the diffusion phenomenon for linear hyperbolic waves.
Convergence to steady states for radially symmetric solutions to a quasilinear degenerate diffusive Hamilton–Jacobi equation
Authors: Barles Guy , Laurençot Philippe , Stinner Christian
Published: November 2019

Convergence to a single steady state is shown for non-negative and radially symmetric solutions to a diffusive Hamilton–Jacobi equation with homogeneous Dirichlet boundary conditions, the diffusion being the p-Laplacian operator, p≥2, and the source term a power of the norm of the gradient of u. As a first step, the radially symmetric and non-increasing stationary solutions are characterized.
Global Strichartz estimates for the wave equation with a time-periodic non-trapping metric
Authors: Kian Yavar
Published: November 2019

We obtain global Strichartz estimates for the solution u of the wave equation ∂t2u−divx(a(t, x)∇xu)=0 with time-periodic metric a(t, x) equal to 1 outside a compact set with respect to x. We assume a(t, x) is a non-trapping perturbation and, moreover, we suppose that there are no resonances zj∈C with |zj|≥1.
Very accurate estimates of the polygamma functions
Authors: Mortici Cristinel
Published: November 2019

The aim of this paper is to give some refinements of an estimation for the polygamma functions, stated in J. Math. Anal. Approx. Theory 1(2) (2006), 124–134. Finally, a general method to establish increasingly accurate estimations is given.
Integral equations in the study of the asymptotic behavior of skew-product flows
Authors: Sasu Adina Luminiţa , Sasu Bogdan
Published: November 2019

The aim of this paper is to present an inedit perspective concerning the study of the asymptotic behavior of variational systems, using distinct techniques compared with those in the existing literature. We give new and complete answers concerning the study of exponential dichotomy of skew-product flows in terms of the solvability of integral equations between Lp-spaces, motivating the techniques by illustrative examples. We present a comparative analysis between the integral admissibility on the real line and on the half-line, pointing out the main differences as well as the advantages of each case.
Dynamic properties of almost monochromatic standing waves
Authors: Elbaz Claude
Published: November 2019

For a scalar field propagating at light velocity c, we show that kinematic and dynamic properties of almost monochromatic standing waves, with frequency Ω(x, t)=ω±δΩ(x, t), where ω is constant, and δΩ(x, t)�ω, are formally identical with mechanical properties of matter. They are both described by equations with the same mathematical structure. The energy conservation stems from stability in time, while the variational principle stems from stability in space. In classical mechanics of a particle, the relativist equations correspond to the geometrical optics approximation as ω→∞. The quantum mechanical equations correspond to the wave optics approximation, in which wave homogeneous Fourier relations are replaced by the material Heisenberg relations.
A remark on the Schrödinger smoothing effect
Authors: Thomann Laurent
Published: November 2019

We prove the equivalence between the smoothing effect for a Schrödinger operator and the decay of the associate spectral projectors. We give two applications to the Schrödinger operator in dimension one.
Stabilization of the linear Kawahara equation with localized damping
Authors: Vasconcellos Carlos F. , da Silva Patricia N.
Published: November 2019

In this erratum of the article: “Stabilization of the linear Kawahara equation with localized damping”, published in Asymptotic Analysis 58(4) (2008), 229–252, DOI 10.3233ASY-2008-0895, we shall fix a mistake which was done when we proved the existence and we defined the countable set of the lengths L where the decay of the energy fails, a set named 𝒩, see Lemma 2.1, p. 235. We considered the equation defined by λu0+u0‴+u0'''''=0, which is wrong since the parameter η in the Kawahara equation was completely forgotten and it makes changing in the result. In fact, η is a negative constant and the correct equation is λu0+u0‴+ηu0'''''=0. Then, as η<0 we can prove, in this case, that the set 𝒩 is empty.

Therefore, we obtain that the energy associated with linear Kawahara equation decays exponentially for all lengths L, even in absence of the damping.

In the new proof for Lemma 2.1, remain the same idea and the same arguments of the original proof, but the signal of parameter η produces different final result.
Asymptotic behavior of solutions to dissipative Korteweg–de Vries equations
Authors: Vento Stéphane
Published: November 2019

We study the large time behavior of solutions to the dissipative Korteweg–de Vries equations ut+uxxx+|D|αu+uux=0 with 0<α<2. We find asymptotic expansions of the solution as t→∞ in various Sobolev norms.
Polarization matrices in anisotropic heterogeneous elasticity
Authors: Nazarov S.A. , Sokolowski J. , Specovius-Neugebauer M.
Published: November 2019

Polarization matrices (or tensors) are generalizations of mathematical objects like the harmonic capacity or the virtual mass tensor. They participate in many asymptotic formulae with broad applications to problems of structural mechanics. In the present paper polarization matrices for anisotropic heterogeneous elastic inclusions are investigated, the ambient anisotropic elastic space is allowed to be inhomogeneous near the inclusion as well. By variational arguments the existence of unique solutions to the corresponding transmission problems is proved. Using results about elliptic problems in domains with a compact complement, polarization matrices can be properly defined in terms of certain coefficients in the asymptotic expansion at infinity of the solution to the homogeneous transmission problem. Representation formulae are derived from which properties like positivity or negativity can be read of directly. Further the behavior of the polarization matrix is investigated under small changes of the interface.
Asymptotical distribution of eigenvalues of dynamics generator governing vibrations of double-walled carbon nanotube model
Authors: Shubov Marianna A. , Rojas-Arenaza Miriam
Published: November 2019

The paper is devoted to rigorous spectral analysis of recently developed mathematical model of a double-walled carbon nanotube. The model is governed by a system of four partial differential equations describing vibrations of two Timoshenko beams coupled through distributed van der Waals forces. The system is equipped with a four-parameter family of nonconservative boundary conditions. The corresponding initial boundary-value problem has been reduced to an evolution equation in the state space. The dynamics generator is a nonselfadjoint matrix differential operator with purely discrete spectrum. It is shown that the entire spectrum asymptotically splits up into four spectral branches. Asymptotical representation has been derived for the eigenvalues along each spectral branch as the number of an eigenvalue tends to infinity. To prove the results, a two-step procedure involving construction of the left and right reflection matrices has been used.
Parameter estimation in hyperbolic multichannel models
Authors: Liu W. , Lototsky S.V.
Published: November 2019

A multichannel model is considered, with each channel represented by a linear second-order stochastic equation with two unknown coefficients. The channels are interpreted as the Fourier coefficients of the solution of a stochastic hyperbolic equation with possibly unbounded damping. The maximum likelihood estimator of the coefficients is constructed using the information from a finite number of channels. Necessary and sufficient conditions are determined for the consistency of the estimator as the number of channels increases, while the observation time and noise intensity remain fixed.
Asymptotic expansion for the widths of resonances in Born–Oppenheimer approximation
Authors: Baklouti Hamadi
Published: November 2019

We develop a microlocal study for scalar classical analytic pseudo-differential operators of principal type. Next, we reduce a system of pseudo-differential operators to the normal form. Then we obtain asymptotic expansion and the exact exponential decay rate for the width of resonances arising from microlocal tunneling effect.
Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations
Authors: Ben Hassen Imen
Published: November 2019

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the nonautonomous first-order problem

u˙+ℳu=g(t),  t∈R+ ,

and for the second-order problem

u¨+u˙+ℳu=g(t),  t∈R+,

where ℳ is a gradient operator and g is a function which tends to 0 when t tends to ∞.
Sharp upper bounds on resonances for perturbations of hyperbolic space
Authors: Borthwick David
Published: November 2019

For certain compactly supported metric and/or potential perturbations of the Laplacian on Hn+1, we establish an upper bound on the resonance counting function with an explicit constant that depends only on the dimension, the radius of the unperturbed region in Hn+1, and the volume of the metric perturbation. This constant is shown to be sharp in the case of scattering by a spherical obstacle.
Rigorous weakly nonlinear geometric optics for surface waves
Authors: Marcou Alice
Published: November 2019

This paper is concerned with surface waves, solutions of hyperbolic nonlinear boundary value problems. We construct BKW solutions in the weakly nonlinear regime with infinite expansion in powers of ε. We rigorously justify this expansion, constructing exact solutions, which admit the asymptotic expansions. We also show that the solution is not necessarily localized at the order 𝒪(ε∞) in the interior, even if the data are.
Liouville theorems for stable radial solutions for the biharmonic operator
Authors: Warnault Guillaume
Published: November 2019

This article is concerned with the non-existence of stable solutions for a fourth-order semilinear elliptic equation Δ2u=f(u) in RN, where f is a smooth nonlinearity. We establish the non-existence of stable radial solutions which verify decay conditions at infinity. Our Liouville-type results do not depend on the specific nonlinearity f. Moreover, in low dimensions and under no radial symmetry assumption, we prove Liouville-type results when f is increasing.
Nonconvergent radial solutions of semilinear elliptic equations
Authors: Kwong Man Kam , Wong Solomon Wai-Him
Published: November 2019

For many known examples of semilinear elliptic equations Δu+f(u)=0 in RN (N>1), a bounded radial solution u(r) converges to a constant as r→∞. Maier, in 1994, constructed, for N=2, an equation with a nonconvergent radial solution. Some necessary conditions for the existence of a nonconvergent solution were given by Maier, and later extended by Iaia. These conditions point out that, for N>2, equations with nonconvergent solutions are rather rare.

A nonconvergent solution must oscillate between two constant values c1<c2 and f must vanish at either c1 or c2. In the neighborhood of one of these points, f must fluctuate wildly in an unusual way that excludes almost all common functions. In this paper, we give a further improvement of the above result with an alternative, simpler proof. The proof depends on an elementary, but nonobvious property of an initial value problem.
Toda system and cluster phase transition layers in an inhomogeneous phase transition model
Authors: Wei Juncheng , Yang Jun
Published: November 2019

We consider the following singularly perturbed elliptic problem

ε2Δũ+(ũ−a(y˜))(1−ũ2)=0 in Ω,  ∂ũ/∂ν=0 on ∂Ω,

where Ω is a bounded domain in R2 with smooth boundary, −1<a(y˜)<1, ε is a small parameter, ν denotes the outward normal of ∂Ω. Assume that Γ={y˜∈Ω: a(y˜)=0} is a simple closed and smooth curve contained in Ω in such a way that Γ separates Ω into two disjoint components Ω+={y˜∈Ω: a(y˜)<0} and Ω−={y˜∈Ω: a(y˜)>0} and ∂a/∂ν0>0 on Γ, where ν0 is the outer normal of Ω+, pointing to the interior of Ω−. For any fixed integer N=2m+1≥3, we will show the existence of a clustered solution uε with N-transition layers near Γ with mutual distance O(ε|log ε|), provided that ε stays away from a discrete set of values at which resonance occurs. Moreover, uε approaches 1 in Ω− and −1 in Ω+. Central to our analysis is the solvability of a Toda system.
Asymptotics of solutions to the Navier–Stokes system with nonzero flux in a layer-like domain
Authors: Pileckas K. , Specovius-Neugebauer M.
Published: November 2019

The asymptotic behavior in a layer-like domain of weak solutions with nonzero fluxes for the Dirichlet problem of the stationary Navier–Stokes system is investigated. If the data decay sufficiently fast, then first three asymptotic terms of any solution which grows at infinity not “too fast” have the same structure as those for the linear Stokes problem. The first asymptotic term which occurs due to the nonlinearity is of order |x|−3 at infinity in the velocity part; it is constructed explicitly. The results are obtained without any assumption on the smallness of the data.
Homogenization of generalized characteristics associated to solutions of Hamilton–Jacobi equations
Authors: Yu Yifeng
Published: November 2019

It was proved in Arch. Ration. Mech. Anal. 162 (2002), 1–23 that singularities of solutions for Hamilton–Jacobi equations will propagate along generalized characteristics. In this paper, we consider the homogenization of generalized characteristics. It is natural to think that (*) generalized characteristics associated to viscosity solutions of oscillatory Hamilton–Jacobi equations will converge to generalized characteristics associated to solutions of the effective equation. We show that this is indeed correct if the spatial dimension is 1. For high dimensions, we need to add some extra assumptions of singularities near generalized characteristics. We provide a counterexample that (*) fails without those assumptions. Some issues related to the weak KAM theory are also discussed.
Small volume expansions for elliptic equations
Authors: Bal Guillaume , Pinaud Olivier
Published: November 2019

This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form ∇·Dε∇uε=0 or (−Δ+qε)uε=0 with prescribed Neumann conditions. The theory is well known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to an order ε2d, where d is dimension and ε is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most εd+1 rather than the expected εd. We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables.
Dimension reduction of a crack evolution problem in a linearly elastic plate
Authors: Freddi Lorenzo , Paroni Roberto , Zanini Chiara
Published: November 2019

A two-dimensional model which describes the evolution of a crack in a plate is deduced from a three-dimensional linearly elastic Griffith's type model. The result is achieved by adopting the framework of energetic solutions for rate-independent processes, to model three-dimensional fracture evolution, in conjunction with a variational dimension reduction procedure.
Asymptotics of characteristic polynomials of Wigner matrices at the edge of the spectrum
Authors: Kösters H.
Published: November 2019

We investigate the asymptotic behaviour of the second-order correlation function of the characteristic polynomial of a Hermitian Wigner matrix at the edge of the spectrum. We show that the suitably rescaled second-order correlation function is asymptotically given by the Airy kernel, thereby generalizing the well-known result for the Gaussian Unitary Ensemble (GUE). Moreover, we obtain similar results for real-symmetric Wigner matrices.
Homogenization of one phase flow in a highly heterogeneous porous medium including a thin layer
Authors: Amaziane B. , Pankratov L. , Prytula V.
Published: November 2019

In this work we consider a model problem describing one phase flow through a thin porous layer made of weakly permeable porous blocks separated by thin fissures. The flow is modeled by a linear parabolic equation considered in a bounded 2D domain with high contrast coefficients. The problem involves three small parameters: the first one characterizes the periodicity of the distribution of the blocks in the layer, the second one stands for the thickness of the layer, the third one characterizes the volume fraction of the fissure part in the layer. Using the notion of two-scale convergence, we derive the homogenized models which govern the global behavior of the flow when the small parameters tend to zero. The global models essentially depend on the relation between the small parameters.
Asymptotic estimates to quenching solutions of heat equations with weighted absorptions
Authors: Wang Wei , Zheng Sining
Published: November 2019

This paper studies heat equations with weighted nonlinear absorptions of the form ut=uxx−Mf(x)u−p in (−1,1)×(0,T) subject to Dirichlet boundary conditions u(−1,t)=u(1,t)=1 and initial data ϕ(x). The asymptotic estimates to quenching time and set of solutions as M→+∞ is established by local energy estimates. It is obtained that the quenching time T~m/(p+1)·M−1 with m=(max x(f(x)/ϕp+1(x)))−1 as M→+∞. It is shown also how the quenching set concentrates near the maximum points of f/ϕp+1 for large M.
High-frequency averaging in semi-classical Hartree-type equations
Authors: Giannoulis Johannes , Mielke Alexander , Sparber Christof
Published: November 2019

We investigate the asymptotic behavior of solutions to semi-classical Schrödinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.
Analysis of a singular limit of boundary conditions for convection–diffusion equations
Authors: Chacón Rebollo Tomás , Gómez Mármol Macarena , Sánchez Muñoz Isabel
Published: November 2019

In this work we justify the way to set the boundary conditions by certain numerical methods to solve convection–diffusion problems, in particular convection–diffusion problems that appear in turbulence models. To do it, we analyze the limit of convection–diffusion equations when the total flux is imposed in the inflow boundary, and Newmann boundary conditions are imposed in the remaining of the boundary. We prove that the solution converges in L2 to the solution of the pure convection problem, with Dirichlet boundary conditions in the inflow boundary, in both the steady and evolution problems. In addition, the convective derivatives also converge in L2, and the convective traces in the inflow and outflow boundaries converge in spaces of L2 kind.
Homogenization of a class of Neumann problems in perforated domains
Authors: Ould-Hammouda Amar
Published: November 2019

We consider elliptic problems in periodically perforated domains in RN, N≥3, with nonhomogeneous Neumann conditions on the boundary of the holes. The aim is to give the asymptotic behavior of the solutions as the period ε goes to zero. Two geometries are considered. In the first one, all the holes are “small”, i.e., their size is of order of εr(ε) with r(ε)→0. The second geometry is more general, there are small holes as before but also holes of size of the order of ε (the last ones corresponding to the classical homogenization situation). Our study is performed by the periodic unfolding method from C. R. Acad. Sci. Paris Ser. I 335 (2002), 99–104, adapted to the case of holes of size εr(ε) (see J. Math. Pures Appl. 89 (2008), 248–277). The use of this method allows us to study second-order operators with highly oscillating coefficients and so, to generalize here the results of RAIRO Modél. Math. Anal. Numér. 4(22) (1988), 561–608. In both cases, if r(ε)=exp (N/N−1), an additional term appears in the right-hand side of the limit equation.
Filtering for fast mean-reverting processes
Authors: Papanicolaou Andrew
Published: November 2019

We consider nonlinear filtering applications to target tracking based on a vector of multi-scaled models where some of the processes are rapidly mean reverting to their local equilibria. We focus attention on target tracking problems because multiple scaled models with fast mean-reversion (FMR) are a simple way to model latency in the response of tracking systems. The main results of this paper show that nonlinear filtering algorithms for multi-scale models with FMR states can be simplified significantly by exploiting the FMR structures, which leads to a simplified Baum–Welch recursion that is of reduced dimension. We implement the simplified algorithms with numerical simulations and discuss their efficiency and robustness.
Semi-classical behaviour of Schrödinger's dynamics: Revivals of wave packets on hyperbolic trajectory
Authors: Lablée Olivier
Published: November 2019

The aim of this paper is to study the semi-classical behaviour of Schrödinger's dynamics for an one-dimensional quantum Hamiltonian with a classical hyperbolic trajectory. As in the regular case (elliptic trajectory), we prove, that for an initial wave packets localized in energy, the dynamics follows the classical motion during short time. This classical motion is periodic and the period Thyp is order of |ln h|. And, for large time, a new period Trev for the quantum dynamics appears: the initial wave packets form again at t=Trev. Moreover, for the time t=p/qTrev a fractional revivals phenomenon of the initial wave packets appears: there is a formation of a finite number of clones of the original wave packet.
Saint-Venant's theory for beams with multi-connected cross-section: Justification and error estimate
Authors: Freddi Lorenzo , Murat François , Paroni Roberto
Published: November 2019

The aim of the paper is twofold. First, starting from the three-dimensional theory of linear elasticity, we give a simple justification of Saint-Venant theory for beams with multi-connected cross-section by means of Γ-convergence. Second, we estimate the error between the three-dimensional problem and the limit problem.
Graph-like models for thin waveguides with Robin boundary conditions
Authors: Cacciapuoti Claudio , Finco Domenico
Published: November 2019

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that in the limit of small width of the waveguide the transverse modes are independent. The projections on each transverse mode generically give rise to decoupling between the edges of the graph while exceptionally a coupling can occur. The coupling takes place if there exists a resonance at the threshold of the continuum spectrum of the effective Hamiltonian resulting from the projection.
Dirichlet boundary control for a parabolic equation with a final observation I: A space–time mixed formulation and penalization
Authors: Ben Belgacem Faker , Bernardi Christine , El Fekih Henda
Published: November 2019

We are interested in the optimal control problem of a parabolic equation with no state constraints, where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. Practical considerations lead us to use Dirichlet controls with no more regularity than square integrability, which arise some technical difficulties in the mathematical analysis. After setting the state equation and the related adjoint equation we fit the two coupled equations into a mixed space–time variational framework. The resulting saddle-point problem turns out to be well posed. We then use a Robin penalization on the Dirichlet control which enables us to re-transcript the mixed problem in a form better suited to numerical computations. We analyze and establish the convergence when the penalty parameter tends to zero, first without additional smoothness assumptions on the optimal control and then for smooth controls.
Higher asymptotics of Laplace's approximation
Authors: Kirwin William D.
Published: November 2019

We present expressions for the coefficients which arise in asymptotic expansions of multiple integrals of Laplace type (the first term of which is known as Laplace's approximation) in terms of asymptotic series of the functions in the integrand. Our most general result assumes no smoothness of the functions of the integrand, but the expressions we obtain contain integrals which may be difficult to evaluate in practice. We then discuss additional assumptions which are sufficient to simplify these integrals, and for the common case that the functions in the integrand admit (finite) Taylor series and the exponent has a nondegenerate minimum, we evaluate the integrals to obtain explicit formulae for the coefficients.
On the interplay of two-scale convergence and translation
Authors: Neukamm Stefan , Stelzig Philipp Emanuel
Published: November 2019

We study the effects of translation on two-scale convergence. Given a two-scale convergent sequence (uε(x))ε with two-scale limit u(x,y), then in general the translated sequence (uε(x+t))ε is no longer two-scale convergent, even though it remains two-scale convergent along suitable subsequences. We prove that any two-scale cluster point of the translated sequence is a translation of the original limit and has the form u(x+t,y+r) where the microscopic translation r belongs to a set that is determined solely by t and the vanishing sequence (ε). Finally, we apply this result to a novel homogenization problem that involves two different coordinate frames and yields a limiting behavior governed by emerging microscopic translations.
Cahn–Hilliard equations with memory and dynamic boundary conditions
Authors: Cavaterra Cecilia , Gal Ciprian G. , Grasselli Maurizio
Published: November 2019

We consider a modified Cahn–Hiliard equation where the velocity of the order parameter u depends on the past history of Δμ, μ being the chemical potential with an additional viscous term αut, α≥0. This type of equation has been proposed by P. Galenko et al. to model phase separation phenomena in special materials (e.g., glasses). In addition, the usual no-flux boundary condition for u is replaced by a nonlinear dynamic boundary condition which accounts for possible interactions with the boundary. The resulting boundary value problem is subject to suitable initial conditions and is reformulated in the so-called past history space. Existence of a variational solution is obtained. Then, in the case α>0, we can also prove uniqueness and construct a strongly continuous semigroup acting on a suitable phase space. We show that the corresponding dynamical system has a (smooth) global attractor as well as an exponential attractor. In the case α=0, we only establish the existence of a trajectory attractor.
An extension of the stochastic two-scale convergence method and application
Authors: Heida Martin
Published: November 2019

The stochastic two-scale convergence method, recently developed in an article by Zhikov and Piatnitskii [Izv. Math. 70(1) (2006), 19–67] is extended to arbitrary probability spaces and is now based on the theory of stochastic geometry instead of random measures. It will be shown that former results on stochastic and periodic two-scale convergence fit into the new approach in a natural way. These results will be applied to functions with jumps on (n−1)-dimensional manifolds, in particular to a homogenization problem of heat transfer through a composite material or polycrystal.
Strong diamagnetism for the ball in three dimensions
Authors: Fournais Søren , Persson Mikael
Published: November 2019

In this paper we give a detailed asymptotic formula for the lowest eigenvalue of the magnetic Neumann Schrödinger operator in the ball in three dimensions with constant magnetic field, as the strength of the magnetic field tends to infinity. This asymptotic formula is used to prove that the eigenvalue is monotonically increasing for large values of the magnetic field.
Heat asymptotics for Dirichlet elliptic operators with non-smooth coefficients
Authors: Miyazaki Yoichi
Published: November 2019

Let A be a 2mth-order elliptic operator in divergence form subject to the Dirichlet boundary conditions in a bounded Cρ domain Ω of Rn, whose coefficients are in Cσ with given σ>0, where ρ=1 if 0<σ≤1 and ρ=σ+1 if σ>1. Then we investigate the asymptotic formula for the partition function, namely the trace of the kernel of exp (−tA) as t→+0, and show that the well-known asymptotic formula for the C∞ case remains valid up to the order according to σ. We also give the asymptotic formula for the heat kernel in Rn, Rn+ and a bounded C1 domain.
Asymptotic eigenfunctions for a class of difference operators
Authors: Klein Markus , Rosenberger Elke
Published: November 2019

We analyze a general class of difference operators Hε=Tε+Vε on ℓ2((εZ)d), where Vε is a one-well potential and ε is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of Hε. These are obtained from eigenfunctions or quasimodes for the operator Hε, acting on L2(Rd), via restriction to the lattice (εZ)d.
Plane Kolmogorov flows and Takens–Bogdanov bifurcation without parameters: The singly reversible case
Authors: Afendikov Andrei , Fiedler Bernold , Liebscher Stefan
Published: November 2019

We consider the Kolmogorov problem of viscous incompressible planar fluid flow under external spatially periodic forcing. Looking for time-independent bounded solutions near the critical Reynolds number, we obtain a dynamical system on a 6-dimensional center manifold. The dynamics is generated by translations in the unbounded spatial direction. Reduction by first integrals yields a 3-dimensional reversible system with a line of equilibria. This line of equilibria is neither induced by symmetries, nor by first integrals. At isolated points, normal hyperbolicity of the line fails due to a transverse double eigenvalue zero. We investigate such a “Takens–Bogdanov bifurcation without parameters” by blow-up and averaging techniques. In particular we describe the complete set ℬ of all small bounded solutions. In the case of a double symmetry of the external force, which leads to a bi-reversible problem, the authors have proved in Asymptot. Anal. 60(3,4) (2008), 185–211, that ℬ consists of periodic profiles, homoclinic pulses and a heteroclinic front–back pair. In the present article we study the more complicated case where only one symmetry is present. Then ℬ consist entirely of trivial equilibria and multipulse heteroclinic pairs. The latter form a very complicated, albeit non-recurrent, set. Graphics of simplest case scenarios for ℬ are included.