bluuebunny/crossref_metadata_2025 huggingface dataset:
Business opportunities in aircraft cabin conversion and refurbishing
Authors: Niţă Mihaela F. , Scholz Dieter
Published: November 2019

This paper identifies several meaningful business opportunity cases in the area of aircraft cabin conversion and refurbishing and predicts the market volume and the world distribution for each of them: 1) international cabins, 2) domestic cabins, 3) aircraft on operating lease, 4) freighter conversions and 5) VIP completions. This implies the determination of cabin modification/conversion scenarios, along with their duration and frequency. Factors driving the cabin conversion and refurbishing are identified. Several aircraft databases, containing the current world feet as well as the forecasted fleet for the next years, are analyzed. The results are obtained by creating a program able to read and analyze the gathered data. It is shown that about 38000 cabin redesigns will be undertaken within the next 20 years. About 2500 conversions from jetliners into freighters and 25000 cabin modifications at VIP standards will emerge on the market. The North American and European markets will keep providing good business opportunities in this area. The Asian market, however, is growing fast, and its very strong influence on demand puts it in the front rank for the next 20 years.
A quantitative exploration of flight prioritisation principles, using new delay costs
Authors: Cook Andrew , Tanner Graham
Published: November 2019

New costs of delay to the airline are presented for a range of aircraft and by phase of flight. It is shown how these costs may be suitable for cost optimisations during flight prioritisation decision-making. Desirable properties of prioritisation procedures are benchmarked against taxation principles. It is shown that Maximum Take-Off Weights, historically established as a charging mechanism in air transport, could actually be used in the context of determining flight prioritisations. Using non-linear delay costs is superior to the use of delay minutes in such optimisations and often produces more equitable solutions. Issues regarding the scale of solutions sought (e.g. by several flights or a whole network), and the implementation of truly dynamic cost optimisation, both require further research.
The heat expansion for systems of integral equations II
Authors: Widom Harold
Published: November 2019

In an earlier work and (n+1)-term “heat expansion” was derived for a class of integral operators on bounded domains of Rn. A certain condition was required without which the expansion could be incorrect. In this paper it is shown that even without this condition n terms of the expansion are correct. In addition, two classes of examples are given where the condition is satisfied.
Homogénéisation du problème de Neumann non homogène dans des ouverts perfores
Authors: Cioranescu Doina , Donato Patrizia
Published: November 2019

We consider the Neumann problem in a domain with periodic holes. The size of the basic cell and of the holes is of order of ε. Some homogenization theorems are proved. There are essentially two limit cases as ε→0. In the first one the mean value of the data on the boundary of the holes is nonzero. This term gives rise to a nonzero contribution in the right-hand side of the homogenized equation. In the second case, when this mean value is zero, the same homogenized equation as for the homogeneous Neumann problem is obtained. General Fourier conditions are also treated.
Large solutions of elliptic equations involving critical exponents
Authors: Atkinson F.V. , Peletier L.A.
Published: November 2019

We consider positive radial solutions of the equation

−Δu=λuq+u(N+2)/(N−2) 1≤q<(N+2)/(N−2)

in the unit ball B in RN, which vanish on the boundary. For c>0, let λ(c) and u(·,c) as denote the eigenvalue and eigenfunction with the property that u(0,c)=c. In this paper we obtain precise estimates for λ(c) and u(·,c) as c tends to infinity.
Skyrmions and symmetry
Authors: Esteban M.J. , Lions P.L.
Published: November 2019

It was already known that when the topological charge k is larger than 1, the ground state energy for Skyrme's problem can never be achieved by a hedge-hog type function. In this paper we prove that in fact the ratio of the ground state energy to the lower bound of the energy for the hedge-hog functions converges to 0 as k−1.
A class of boundary homogenization problems for parabolic equations
Authors: Tatsien Li , Qi Tang
Published: November 2019

In this paper we present and discuss a class of problems of boundary homogenization for parabolic equations, and point out its application to the heat analysis of a car wheel.
Comportement semi-classique du spectre conjoint d'opérateurs pseudodifférentiels qui commutent
Authors: Charbonnel Anne-Marie
Published: November 2019

We consider ν pseudodifferential operators, Q1(h),…,Qν(h), acting in Rn, commuting together, depending on a small parameter h. For instance, one of them is the Schrödinger operator with a potential V(x) satisfying the condition $\underline{lim}_{|x|\rightarrow +\infty}V(x)>E$ . Under suitable conditions, we establish a functional ca1culus to define f(Q1(h),…,Qν(h)) as a pseudodifferential operator of the same type, when f belongs to C∞0(Rν). We use it to study the semiclassical behaviour of the joint spectrum of Q1(h),…,Qν(h), lying in a compact of Rν where it is discrete. When these operators form a quantically integrable system, we give more precise estimations. Our results generalize those obtained by Helffer and Robert for one operator acting in Rn.
G-convergence of cyclically monotone operators
Authors: Defranceschi Anneliese
Published: November 2019

The paper deals with equations of the form −div(a(x,Du))=f, where a(x,ξ) is cyclically monotone. The main result is that, given a sequence ah of maximal monotone operators of this type, the convergence of the sequence of the solutions uh to the equations −div(ah(x,Duh))=fh implies the convergence of the corresponding momenta ah(x,Duh) in spite of the fact that the functions ah are nonlinear. The equivalence between the Γ-convergence of certain integral functionals and the G-convergence of the corresponding sub differentials is also established.
Estimates for elliptic singular perturbations in Lp-spaces
Authors: Sweers G.
Published: November 2019

In this paper we study two-sided a-priori estimates in Lp-type spaces for a class of singularly perturbed elliptic problems with Dirichlet boundary conditions.
Justification of the boundary conditions of a clamped plate by an asymptotic analysis
Authors: Ciarlet Philippe G. , Le Dret Hervé
Published: November 2019

On considère un problème d'élasticité linéarisée tridimensionnelle, posé sur un domaine constitué d'une plaque d'épaisseur 2ε, insérée dans un support tridimensionnel élastique. En supposant que les constantes de Lamé du matériau constituant la plaque varient comme ε−3 et que celles du support varient comme ε−2−s, s>0, on établit la convergence en norme H1, lorsque ε tend vers zéro, des composantes (convenablement mises à l'échelle) du champ de déplacements vers la solution du problème aux limites bien connu pour une plaque encastrée. De la sorte, on montre que les conditions aux limites classiques pour l'encastrement d'une plaque peuvent être obtenus de façon rigoureuse par une “analyse limite” qui prend également en compte le caractère élastique du support.
Reduction of linear differential systems to Arnold canonical form
Authors: Stengle Gilbert
Published: November 2019

We consider systems of linear differential equations of the form εPdy/dx=A(k,ε)y where A is an n×n matrix-valued function holomorphic in x possessing an asymptotic power series expansion in ε, A(k,ε)≈ΣkAk(x)εk uniformly valid on a neighborhood |x|<r of x=0 as ε→0+. We show that any such system can be reduced by a change of variable y→Ty to a certain generic canonical form depending on the Jordan canonical form of A(0, 0). Here T(x, ε) is an invertible matrix-valued function similarly possessing an asymptotic expansion T≈ΣkTk(x)εk uniformly valid on an ε-dependent domain |x|<rδ(ε) the size of which is determined by certain finer properties of the asymptotic expansion of A. We also obtain new results of classical type in which this domain is independent of ε. The method of proof seems both new and of general purpose. It is based on an exacting analysis of finer termwise properties of divergent infinite formal series transformations which shows certain near-optimal finite sections to be powerful first approximations to actual transformations.
The spectrum of an ideal linear magnetohydrodynamic model with translational symmetry
Authors: Raikov G.D.
Published: November 2019

We consider an ideal linear magnetohydrodynamic model with translational symmetry and investigate the spectral properties of the force operator Ak with a fixed longitudinal wavenumber k. We establish that the essential spectrum of Ak consists of two bounded segments (slow magnetosonic and Alfvén continuum). Next, we show that the isolated eigenvalues of Ak do not accumulate to the tips of the Alfvén continuum. Further, we obtain detailed information about the asymptotics of the eigenvalues of Ak which lie to the right of the essential spectrum and tend to +∞. At last, in the case when the two segments of the essential spectrum of Ak are disjoint, we find that, generically, the isolated eigenvalues of Ak accumulate to the right tip of the slow magnetosonic continuum, and calculate the first asymptotic term of the corresponding infinite eigenvalue sequence.
Justification of elastoplastic plate models
Authors: Qi Tang
Published: November 2019

Starting with a three-dimensional Hencky model on an open set ω×]−ε,ε[, we give a mathematical justification of two-dimensional elastoplastic plate models. Following the work of Duvaut, Lions and Temam, we use the variational form of elastoplastic problems as the starting point of discussion and we use the theory of Γ-convergence to examine the limiting behaviour of the problem when ε→0. We conclude that the solutions of three-dimensional plate problems converge to those of two-dimensional plate problems under regularity hypotheses on the boundary conditions. Finally, convergence of the limit analysis energy is also verified.
Diffusion approximation for a Knudsen gas in a thin domain with accommodation on the boundary
Authors: Babovsky Hans , Bardos Claude , Platkowski Tadeusz
Published: November 2019

The purpose of this work is to show how a rough boundary described by a diffusive boundary condition may generate a diffusion process in a fluid. At variance with previous results obtained by probalistic argument (cf. Babovsky (1986)), our proof relies entirely on functional analysis. It may be less intuitive but it is convenient for generalization to related phenomena. It is based on the introduction of a small parameter ε which describes the thickness of the domain compared with the effect of the boundary. Since we are aiming at a diffusion process it is also natural to introduce a scaling in time. This scaling will also be chosen of the order of 1/ε.
Asymptotique à grande energie de la phase de diffusion pour un potentiel
Authors: Robert Didier
Published: November 2019

About ten years ago several people, Buslaev, Colin de Verdière, Guillopé, and Popov, proved a complete asymptotic expansion, as the energy λ goes to +∞, for the scattering phase s(λ) associated with the Schrödinger operator H=−½Δ+V for compact support, smooth potentials V on Rn. In the present paper, using others techniques, we extend this result to smooth potentials V such that its derivatives of order α are O(|x|−p−|α|) for some p>n. For V itself this decreasing assumption is more or less optimal.
Decay estimates for the solutions of the system of crystal optics
Authors: Liess Otto
Published: November 2019

In the present paper, we study decay estimates for solutions of Maxwell's system in homogeneous anisotropic media. While the estimates are similar to related estimates for the wave equation, the main novelty comes from the fact that here we have a case of characteristics with variable multiplicity.
An asymptotic analysis of the one-dimensional Vlasov–Poisson system: the Child–Langmuir law
Authors: Degond P. , Raviart P.A.
Published: November 2019

We perform the asymptotic analysis of the one-dimensional Vlasov–Poisson system when singular boundary data are prescribed. Such a singular perturbation problem arises in the modelling of vacuum diodes under very large applied bias, and gives rise to the well-known “Child-Langmuir law”. In this paper, we provide a mathematical framework to this physical theory, by successively investigating the reduced problem (when the perturbation parameter ε is set equal to zero) and the boundary layer problem, which gives a sharp qualitative information.
Lateral connection problem and Stokes phenomenon for certain functional spaces
Authors: Tovbis A.I.
Published: November 2019

Our purpose is to investigate a lateral connection problem for matrix differential equations with an irregular singularity in case of a multiple Stokes ray. We reduce this problem to constructing the so-called Stokes operators in certain functional spaces using the results of [7,9]. We also extend the recently obtained [5,10] analytical expression for Stokes matrix in case of a multiple Stokes ray.
Convergence of approximation schemes for fully nonlinear second order equations
Authors: Barles G. , Souganidis P.E.
Published: November 2019

We present a simple, purely analytic method for proving the convergence of a wide class of approximation schemes to the solution of fully non linear second-order elliptic or parabolic PDE. Roughly speaking, we prove that any monotone, stable and consistent scheme converges to the correct solution provided that there exists a comparison principle for the limiting equation. This method is based on the notion of viscosity solution of Crandall and Lions and it gives completely new results concerning the convergence of numerical schemes for stochastic differential games.
Junctions between three-dimensional and two-dimensional nonlinearly elastic structures
Authors: Aufranc Martial
Published: November 2019

A mathematical modeling of the junction between three-dimensional and two-dimensional linearly elastic structures has been recently proposed by P.G. Ciarlet, H. le Dret and R. Nzengwa. Their approach is generalized here to nonlinear elastic structures.
Multisummability of formal power series solutions of linear ordinary differential equations
Authors: Balser W. , Braaksma B.L.J. , Ramis J.-P. , Sibuya Y.
Published: November 2019

If f is a formal power series solution of a linear meromorphic differential equation, then it is shown that f=f1+…+fq, were fj is hj-summable; here h1,…hq are slopes of the associated Newton polygon. The multisummability properties of fundamental solutions of linear homogeneous meromorphic differential equations are studied.
Existence and regularity of functions which minimize certain energies in homotopy classes of mappings
Authors: Duzaar Frank , Fuchs Martin
Published: November 2019

Given a continuous mapping φ:M→N of compact manifolds we prove that there exists a differentiable solution of the problem Ep(u):=∫M|du|pdv $\leadsto$ Min in the homotopy class [φ] provided the sectional curvature of N is nonpositive. Here p∈[2,∞) is an arbitrary real number, hence we obtain an extension of a well-known theorem of Eells and Sampson. Our arguments are based on an asymptotic analysis of the less degenerate problems (0<ε<1) Epε(u):=∫M(ε+|du|2)p/2dv $\leadsto$ min in [φ] for which we establish the existence of solutions uε which converge to a minimizer of our original variational problem. As further applications the paper contains various estimates for p-harmonic maps into negatively curved target manifolds.
Hyperbolic geometric asymptotics
Authors: Alber Mark S.
Published: November 2019

Geometric asymptotics in hyperbolic case are investigated. In particular, semiclassical solutions (modes) of the stationary Schrödinger equation corresponding to the Jacobi problem on geodesics on n-dimensional hyperboloids are constructed in the form of the multi-valued functions of several complex variables (geometric asymptotics) on the Jacobi varieties. Then a special limiting process (collapsing construction) is used in order to obtain from this problem reflected and diffracted modes for investigating geometric theory of diffraction.
A new type of singularly perturbed diffusion and its applications
Authors: Peng Shige
Published: November 2019

A new type of singularly perturbed stochastic differential equation is discussed. Its limit problem, which seems ill-posed, is proved to be well-posed under a certain reasonable condition. We show also how to apply the above result to treat a simulated annealing problem.
Approximation of solutions of y″=ϕ(x,ε)y with several transition points and moving singularities
Authors: Gingold H.
Published: November 2019

Given a second-order linear differential equation y″=ϕ(x,ε)y. Let ϕ(x,ε) be a meromorphic real-valued function of two independent variables x and ε. x is allowed to vary on an interval [a,b] and ε varies on (0,ε0]. The approximation of its solutions, as well as of their derivatives as ε→0+ are provided for the entire interval [a,b] in the presence of several-coalescing transition points. Approximations on closed subintervals of [a,b] are provided. The end points of those subintervals are allowed to coincide with turning points.

The approximations given could replace a nonrigorous analysis in the literature which had similar goals and in which an error was found. The formulas could prove to be useful for numerical implementation in problems of wave propagation in inhomogeneous medium.

Our methods avoid analytic continuation, Langer transformations and special functions.
Asymptotic boundary conditions for an oxide region in a semiconductor device
Authors: Gamba Irene Martínez
Published: November 2019

We study how the equations of the potential of a semiconductor device, as described by Markowich (1986) and Selberherr (1984), behave when the width and the permittivity of the oxide region go to zero. According to their ratio, the problems converge to different asymptotic limits. This provides the correct boundary value approximation to the full problem avoiding costly computations (see, as an example, the discussion in Chapter 5 of (Selberherr, 1984), about the boundary conditions).
On a bending and torsion asymptotic theory for linear nonhomogeneous anisotropic elastic rods
Authors: Alvarez-Dios J.A. , Viano J.M.
Published: November 2019

We use asymptotic techniques on a re-scaled three-dimensional linear elasticity model to justify and generalise models describing the elastic behaviour of rods, taking into account the nonhomogeneity and anisotropy of the material they are made of. General bending models for nonhomogeneous anisotropic rods are obtained as the limit of the tridimensional elasticity model when the cross section area tends to zero. We also include convergence results mathematically justifying the method. Next, by identification of second-order terms in the asymptotic expansion we obtain a general theory for transversally nonhomogeneous isotropic rods taking into account Timoshenko and Saint Venant's effects.
Modeling of the junction between a plate and a rod in nonlinear elasticity
Authors: Gruais Isabelle
Published: November 2019

The asymptotic expansions method is used to study the limit behaviour of a system made of a plate of small thickness 2ε in which one partly inserted a rod of cross section equal to 4ε2. When the free extremity of the rod is clamped and when the materials are nonlinearly elastic, it turns out that the transverse displacement of the rod around the junction is “almost” rigid. Moreover, the plate, which is somehow held by the rod, has a torsion equal to zero around the junction, which makes impossible any free rotation around the axis of the rod.
Summability of asymptotic power series
Authors: Jurkat Wolfgang B.
Published: November 2019

New summability methods, similar to Lindelöf's, are introduced which sum “most” asymptotic power series which occur in applications.
Elliptic perturbations for a class of Hamilton–Jacobi equations
Authors: Eizenberg A.
Published: November 2019

A Hamilton–Jacobi equation, with a convex Hamiltonian such that the corresponding Langrangean vanishes along the orbits of certain dynamical systems, is considered. Small elliptic and inhomogeneous perturbations of such HJE are studied in bounded domains. In the problem at hand the HJE may have more than one viscosity solution under zero boundary conditions, which brings up the question of uniqueness conditions. It is found that under the uniqueness conditions the problem can be reduced to the linear case by a completely new method (notice that the logarithmic transformation, generally speaking, cannot be applied in the considered problem).
Global existence of solutions of second order differential systems
Authors: Cantarelli Giancarlo
Published: November 2019

The methods introduced in [3–5], that is comparison method together with the use of suitable Liapunov functions, are generalized and applied to the study of global existence of solutions of ‘generalized’ Lagrange equations. New criteria are obtained which extend various results of [6] and [7].
Perturbed bifurcation in the von Kármán equations
Authors: Quintela Estévez P.
Published: November 2019

An asymptotic method is presented to analyse perturbations of bifurcations of the solutions of nonlinear problems. The perturbations may result from imperfections, impurities or other inhomogeneities in the corresponding physical problem. Using a method of matched asymptotic expansions we obtain global representations of the solutions of the perturbed problem when the bifurcation solutions are known globally. Even if, in this paper, the asymptotic method is used to analyse the perturbed bifurcation in the von Kármán equations, the same analysis is also valid to study the perturbations in more general nonlinear problems.
Asymptotically periodic solutions of the wave equation in the Hilbert space
Authors: Eidus D.
Published: November 2019

The large time asymptotics of solutions of the Cauchy problem for a non-homogeneous wave equation with a periodic perturbation is found under some restrictions, imposed on the corresponding operator. We apply the obtained results to the third boundary value problem for the wave equation in a half-space.
On the existence of eigenvalues of a divergence-form operator A+λB in a gap of σ(A)
Authors: Alama S. , Avellaneda M. , Deift P.A. , Hempel R.
Published: November 2019

We consider uniformly elliptic divergence type operators A=−Σ∂jaij(x)∂i with bounded, Lipschitz continuous coefficients, acting in the Hilbert space L2(Rν). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study the family of operators

A+λB=−Σ∂j(aij(x)+λbij(x))∂i, λ≥0,

where A is supposed to have a spectral gap, while (bij)≥0 and bij(x)→0, as x→∞. One of our main results assures that discrete eigenvalues of A+λB move into the gap, as λ increases, if the support of the matrix function (bij) is large enough. In addition, we analyze the connection between decay properties of the coefficient matrix (bij) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrödinger case.
Eigenvalue distribution of elliptic operators of second order in domains with infinitely many hooks
Authors: Berger Günter
Published: November 2019

Spectral properties of strongly elliptic operators of second order on bounded domains with infinitely many hooks and Neumann boundary conditions are considered. In the case of a pure point spectrum asymptotic formulae with remainder estimates for the counting function N(λ) of the eigenvalues are proved. These formulae yield the conclusion that the asymptotic eigenvalue distributions coincide with the classical formula of Weyl.
Non-convergent radial solutions of semilinear elliptic equations
Authors: Maier Stanislaus
Published: November 2019

We study solutions u=u(t) of an initial value problem for u″+(n−1)/t u′+f(u)=0, which have the additional property that the limit of u as t approaches infinity does not exist. Besides some examples, we give necessary conditions on the non-linearity f for the existence of such (non-convergent) solutions. One corollary of these investigations will be that, if f(u)u>0 for small |u|≠0 then every solution u(·;p) (with |p| small enough) of the initial value problem (1.1), stated below, converges to zero as t→∞.
A generalized Cantor–Riesz–Nagy function and the growth of its moments
Authors: Goh William M.Y. , Wimp Jet
Published: November 2019

In this paper we introduce a singular function which reflects some of the behavior of both the classical Cantor singular function and of the Riesz–Nagy class of singular functions. We determine the asymptotic behavior of the corresponding moment sequence. The behavior of the moments is similar to the behavior of the moments of two singular functions we have considered previously.
Existence of solutions for a homogenized hyperbolic system of conservation laws
Authors: Heibig A.
Published: November 2019

This paper deals with solutions of a homogenized hyperbolic system of conservation laws. Recently, Weinan E (case of a one-dimensional compressible fluid, cf. [3]) and D. Serre (general case of a one-dimensional system of conservation laws, cf. [11]) derived a system of homogenized equations, by using a multiple scale analysis. Our goal is to prove the existence of solutions for such a system.
Dimension reduction in variational problems, asymptotic development in Γ-convergence and thin structures in elasticity
Authors: Anzellotti G. , Baldo S. , Percivale D.
Published: November 2019

We consider families of variational problems Fε over domains Ωε whose extension in one or more directions is small compared to the extension in the other directions, and goes to zero while ε tends to zero. We study then the “variational” convergence of the functionals Fε to a new functional defined on a domain A in a lower dimensional space, where those “dimensions” that were small in Ωε disappear. A general framework is presented in the first part of the paper and an application to the elastic rod and the elastic plate is given in the second part.
On passing to the limit for non-convex variational problems
Authors: Amar Micol , Cellina Arrigo
Published: November 2019

In this paper, we consider a sequence of integral functionals Fn:X→(−∞,+∞], where X is the set of those functions u belonging to W1,p(0, T), p > 1, satisfying: u(0) = A, u(T) = B. For every n∈N,Fn is represented by the sum of two integrands, where the first one is T-periodic in time and non-convex with respect to u′ and the second one depends only on u.

We give a necessary and sufficient condition in order to obtain the existence of an integral functional F∞:X→(−∞,+∞] such that, for every minimizing sequence (un) converging to u∞, the lower limit of the corresponding sequence Fn(un) coincides with F∞(u∞). The integrand function in F∞ does not depend on time and, in general, it is non-convex with respect to u′.
Necking deformation in non-linear elasticity
Authors: Mascolo E. , Migliaccio L.
Published: November 2019

We study the asymptotic behaviour of the minimizers of the variational problem:

min {∫a−a[ε2(u′)2(x)+W(u(x))]dx:u∈L1(−a,a),u≥0,∫a−au(x)dx=m,u(−a)=u(a)=c},

where m/2a∈(α,β) and W is a non-negative, continuous and non-convex function, with W(u) = 0 iff u∈{α,β}.

We prove that the presence of the necking is related to the value c; more precisely, we have a “neck” if c<α and a “buldge” if c>β.
Wave propagation in even-dimensional spaces
Authors: Ozawa T.
Published: November 2019

Asymptotic expansions of solutions of the wave equations in even dimensional spaces are obtained with the initial data of non-compact support. A relationship is proved between the vanishing order at the origin of the Fourier transform of the data and the decay rate of the corresponding solutions in semi-infinite cylinders or along rays inside the forward light cone.
Inertial effects in the asymptotic behavior of a thin film flow
Authors: Assemien Ahiko , Bayada Guy , Chambat Michèle
Published: November 2019

We study the dependence with respect to the domain of the constants appearing in some Sobolev embeddings. It shows that the inertial effects are negligible at the first order for the asymptotic behavior of a thin film flow. The influence of the inertial effects is obtained at the second order in the asymptotic expansions of the pressure and velocity, with full convergence argument for chosen boundary conditions.
Isolated singularities and asymptotic behaviour of the solutions of a semi-linear heat equation
Authors: Moutoussamy I. , Veron L.
Published: November 2019

We study here the limit properties of the solutions of the following singular problem (1) ut−Δu+u|u|p−l=0 in RN×R+\{(0,0)} with u(x,0)=0 if x≠(0,0). We transform the equation from an autonomous dynamical system into a weighted Sobolev space and look for the limit set of a trajectory of such a system. By using an energy method, we obtain a theorem of classification of singularities of the solutions of (1). By using the same methods we study the asymptotic behaviour of the solutions and some global singular solutions.
Homogenization of non-linear random media by a duality method. Application to plasticity
Authors: Sab K.
Published: November 2019

The homogenization of non-linear elastic media and elasto-piastic media is investigated. It is shown that the results established for periodic media hold also for statistically homogeneous ergodic (S.H.E.) media. The homogenization scheme is based on (i) — a formal analogy between periodic media and S.H.E. media, and, (ii) — the study of the duality in S.H.E. non-linear media. “Explicit” direct and dual definitions of the homogenized properties are provided and illustrated by an example.
Homogenization of two-phase immiscible flows in a one-dimensional porous medium
Authors: Bourgeat Alain , Mikelić Andro
Published: November 2019

We consider the hyperbolic-elliptic coupled nonlinear system describing two-phase immiscible flows, with neglected capillary pressure, through a one-dimensional porous medium. We prove the existence and uniqueness of an entropy solution for data corresponding to the solution with discontinuities in saturations. Then we study the homogenization of this system when the permeability and porosity are rapidly oscillating. We get the stability of our coupled system, due to the very special underlying structure of this system in one dimension. Finally we discuss the behavior of the system with oscillatory initial saturation. This result also contains the homogenization of two-component miscible flow in a one-dimensional porous medium, without any dispersion term, as a special case.
Analysis and asymptotics of a one-dimensional ion extraction model
Authors: Abdallah Naoufel Ben , Mas-Gallic Sylvie , Raviart Pierre-Arnaud
Published: November 2019

In this paper a one-dimensional model for ion extraction from a plasma is investigated. The analysis is achieved thanks to a singular perturbation of the transport equations based on the smallness of both, the Debye length and the thermal voltage. The convergence of the perturbed problems to a limit problem, where a “free surface” appears, is proved (in one dimension the free surface reduces to a single point). The analysis is first done for finite temperature ions. For cold ions, the analysis can be carried out further: a boundary layer lying at the free surface is exhibited and analyzed.
On a singular diffusion equation originating from porous media flow
Authors: van Duijn C.J. , Floris F.J.T.
Published: November 2019

In this paper we consider the singular boundary value problem, −z1/pz″=D(f)/(1+p) for 0<f<1 with z(0)=z(1)=0. Here p>0 and D: [0,1]→R is a given function. This problem arises in a model for two-phase capillary induced flow in porous media. Considering the special case D(f)=fαw(1−f)α0, with α0,αw>−2, we investigate the singular behaviour of the solution z(f) as α0,αw↓−2. We show that the solution then becomes unbounded. We investigate the behaviour of z and z′ in this limit process. The results are incorporated in an algorithm which we use to solve the problem numerically. The numerical results show significant improvement over standard discretisation techniques near the limit. Non-existence arises for α0 or αw≤−2.
Asymptotics of variational eigenvalues of non-linear Sturm-Liouville problems with two parameters on general level sets
Authors: Shibata Tetsutaro
Published: November 2019

We consider the non-linear Sturm-Liouville problem with two parameters on general level set Nα: [Formula: see text] where p>1, μ,λ∈R are parameters and [Formula: see text] We establish an asymptotic formula of n-th variational eigenvalue λ=λn(α) of (1) as α→−∞:

λn(α)=Cμ(−α)(1−p)/2+o((−α)(1−p)/2).

Furthermore, we give an asymptotic formula of Cμ as μ→∞.
Fonction de coût et contrôle des solutions des équations hyperboliques
Authors: Robbiano L.
Published: November 2019

Dans cet article, on donne une estimation de l'énergie du contrôle de la solution d'un problème hyperbolique. Cette estimation est de la forme eC/ε2 si ε est le maximum de la norme de la solution à l'instant T.
Double obstacle formulation with variable relaxation parameter for smooth geometric front evolutions: asymptotic interface error estimates
Authors: Nochetto R.H. , Paolini M. , Verdi C.
Published: November 2019

A singularly perturbed double obstacle problem is examined as a variational tool for the approximation of the geometric motion of fronts. The relaxation parameter is space-time dependent, thereby allowing the control of transition layer thickness and related interface pointwise accuracy. Optimal order interface error estimates are derived for smooth evolutions. The estimates have a local character for small time, namely they depend on the relaxation parameter local magnitude. The proof is based on constructing suitable sub and supersolutions, which incorporate a number of shape corrections to the basic standing wave profile, and using a modified distance function to the front. Numerical simulations illustrate how the variable transition layer thickness can be exploited in dealing with large curvatures and ultimately in resolving singularities.
Effet d'Aharonov-Bohm sur un état borné de l'opérateur de Dirac
Authors: Parisse B.
Published: November 2019

On s'intéresse ici à l'étude lorsque h→0 (en limite semi-classique) de l'opérateur de Dirac défini sur C∞(R3;C4) par:

(1) Dt(h)=Σ3j=1αj(hDj+tAj)+α4+V(x).I4,

en présence d'un champ magnétique B = rot A nul en-dehors d'un cylindre (ici les αj désignent des matrices 4 × 4 qui anticommutent). Dans le cas de l'opérateur de Schrödinger en dimension 2, si le champ magnétique est nul en-dehors d'un disque, B. Helffer a montré qu'il peut apparaître un effet de flux qui décale les valeurs propres de l'opérateur sans champ magnétique, cet effet de flux est nommé effet d'Aharonov-Bohm. On montre ici qu'il en est de même pour l'opérateur de Dirac et on verra qu'en limite non-relativiste l'effet d'Aharonov-Bohm provoque un découplage des valeurs propres (qui sont de multiplicité paire en l'absence de champ magnétique).
Comportement asymptotique de solutions d'équations elliptiques semi-linéares dans un cylindre
Authors: Brada A.
Published: November 2019

Cet article a pour but l'étude du comportement asymptotique de la solution d'équations d'évolution du second ordre, de la forme: [Formula: see text] Ω étant, soit un ouvert borné de RN (partie I et II), soit RN (partie III), et β une fonction continue, monotone, sur R.

De nombreux résultats ont été obtenus ces dernières années sur de tels sujets, notamment par H. Bérestycki et L. Nirenberg dans [4], puis par V.A. Kondratiev et O.A. Oleinik (voir [11] et [12]).

Dans [4], H. Bérestycki et L. Niremberg étudient, entre autres, des équations du type [Formula: see text] où S={x=(x1,x2,…,xn); y=(x2,…,xn)∈Ω}, Ω étant un domaine borné de Rn−1.

Dénotant par [Formula: see text] (avec g(y,0)=0), et désignant par λ1 la première valeur propre de l'opérateur (−Δ+a(y)) dans Ω, avec condition de Dirichlet sur ∂Ω, ces auteurs montrent que pour β=0, et sous l'hypothèse λ1>0 (et fonction propre correspondante ϕ>0), il existe une constante positive α telle que la solution u de (2) avec condition de Dirichlet u=0 sur ∂S, vérifie: [Formula: see text] lorsque x1→+∞.

Des résultats analogues sont aussi prouvés lorsque des conditions de Riemann sont imposées sur ∂S.

Les résultats que nous proposons ici sont, pour certains, antérieurs aux travaux cités ci-dessus (voir [5]): ils font suite à ceux obtenus par L. Véron et P. Baras dans [2], et L. Véron et A. Gmira dans [9] et [10], pour l'équation parabolique: [Formula: see text] Ce travail comporte trois parties:

1. Comportement asymptotique de la solution u de (1) dans un ouvert borné avec condition de Neumann au bord. Plus particulièrement, pour $\beta{:}\ r\mapsto r|r|^{p-1},p>1$ , et u0∈L1(Ω), on obtient: [Formula: see text] uniformément sur $\overline{\Omega}$ .

On peut, en outre, préciser ce qui se passe si l=0: il existe un entier k>0 telle que $\mathrm{e}^{\sqrt{\lambda_{k}}t}u(t,\cdot)$ converge, pour t→+∞, vers un élément non nul ψk(·)∈ker(Δ+λk id) (λk étant une valeur propre de −Δ dans Ω). 2. Comportement pour des conditions de Dirichlet au bord.

Pour u0∈L2(Ω), et en désignant par λ1 la première valeur propre de −Δ dans Ω, on montre que [Formula: see text] Cette limite est donc de la forme αh0, h0 étant un générateur du sous-espace propre: on étudiera alors quelques propriétés de l'application $u_{0}\mapsto \alpha$ .

En particulier, pour β(r)=r|r|p−1 et α=0, il existe une valeur propre λk de −Δ dans Ω, telle que [Formula: see text]

3. Comportement asymptotique dans RN.

Ici aussi, il dépend essentiellement du signe de p−(N+2)/N et de la décroissance de u0. On aura: [Formula: see text] pour u0≥0 et [Formula: see text] Pour u0∈L1(RN) et p≥(N+2)/N, on obtient: [Formula: see text] uniformément sur les ensembles de la forme {x∈RN: |x|≤ct}. En outre, si p=(N+2)/N, C=0.
Convergence of displacements and stresses in linearly elastic slender rods as the thickness goes to zero
Authors: Le Dret Hervé
Published: November 2019

We show the convergence in an appropriate sense of displacements and stresses in a linearly elastic slender rod when its thickness tends to zero. The limit displacements and stresses are determined by well-posed variational problems.
Reactive flows in layered porous media, I. Homogenization of free boundary problems
Authors: Xin J. , Peirce A. , Chadam J. , Ortoleva P.
Published: November 2019

A model of reactive flow in a layered porous medium is considered in which the layering is represented by small-scale periodic structure. A novel form of homogenization analysis is presented, combining geometric optics and multiple scales expansions together with matched asymptotics to derive an effective free boundary problem for the motion of the reactive interface. Applications of the effective free boundary equations are given in which travelling wave solutions and the stability of shape perturbations are considered.
An existence theorem for weak solutions of the semiconductor equations modelling the non-stationary avalanche generation
Authors: Naumann J.
Published: November 2019

The paper is concerned with the non-stationary semiconductor equations (van Roosbroeck's system) involving a gradient non-linearity which models the generation of particles due to impact ionization. We establish the existence of a weak solution to this system over a cylinder Ω×(0,T1) (Ω⊂R2 bounded domain, (0, T1) small time interval). The method of proof relies on approximating the generation term by bounded gradient non-linearities.
Bounds for the dimensions of the attractors of non-linear bipolar viscous fluids
Authors: Bellout Hamid , Bloom Frederick , Nečas Jindrich
Published: November 2019

The existence of a maximal global attractor is demonstrated for the dynamical system generated by the initial boundary value problem associated with the motion of a non-linear bipolar fluid; the attractor has the form A=∩t>0S(t)Bρ′(H2(Ω))3 where S(t) is the relevant non-linear semigroup and Bρ′(H2(Ω))3 (the ball of radius ρ′>0 in (H2(Ω))3) is, for ρ′ sufficiently large, an absorbing set. Upper bounds are obtained for the Hausdorff and Fractal dimensions of the attractor A.
On the Hausdorff dimension of n×m concentration sets
Authors: Bildhauer Michael
Published: November 2019

The Hausdorff dimension of n×m concentration sets is studied. Uniform dimension estimates at each cross section in Rn and Hölder continuity in Rm are used to deduce dimension estimates in Rn+m.
Effective fluid flow in a porous medium containing a thin fissure
Authors: Bourgeat Alain , Marušić-Paloka Eduard , Mikelić Andro
Published: November 2019

We consider the behavior of the solution to Stokes system in porous medium, with characteristic size of the pores ε and containing a thin fissure {0≤xn≤η}, of width η. The limit when ε and η tend to zero gives the effective (or homogenized) behavior for the flow. Depending on the comparison between ε and η we obtain three possible limits:

– When η<κε2/3 the fissure is not giving any contribution, i.e., the effective fluid flow is governed by the Darcy law.

– When η~κε2/3 the coupling effect appears and the effective flow behaves like Darcy flow in the porous medium coupled with the tangential flow on the surface {xn = 0}. Compared to the previous case η<ε2/3, the effective velocity has now an additional tangential component concentrated on {xn = 0}.

– When η>κε2/3 the fissure is dominant and the tangential flow at {xn = 0} is dominant. Meanwhile in the other part of the porous medium the effective velocity is equal to zero.
An inverse problem originating from magnetohydrodynamics, III. Domains with corners of arbitrary angles
Authors: Beretta Elena , Vogelius Michael
Published: November 2019

We seek to identify the non-linearity of the semilinear elliptic equation, ▵u=−f(u)≤0, from boundary measurements of the normal flux corresponding to homogeneous Dirichlet data. The possibility of such identification depends crucially on the shape of the domain. In this paper we prove that identification of an analytic function f is (generically) possible if the domain has a proper corner. This result significantly extends an earlier result obtained in [2], by almost entirely eliminating the restrictions imposed on the size of the angle of the corner.
Oscillations de faible amplitude pour les systèmes 2×2 de lois de conservation
Authors: Cheverry C.
Published: November 2019

Cet article est consacré aux systèmes de deux lois de conservation. Les conditions initiales sont choisies sous la forme d'oscillations de faible amplitude. On prouve que les solutions BV obtenues à l'aide du schéma de Glimm convergent vers l'unique solution entropique des équations de modulation de l'optique géométrique non linéaire.
The ‘breather-train’ solution of the Sine-Gordon equation with the ‘periodic wave–zero’ boundary conditions
Authors: Novokshenov V.Yu.
Published: November 2019

The large-time asymptotic solution of the Sine-Gordon equation is considered describing the decay of a step-like initial condition with nonidentical finite-gap limits as x→±∞. The leading term of asymptotics is found as a finite-gap quasi periodic solution with its phase vectors modulated by a slow space-like variable. The Whitham equations describing the modulation are studied in detail for the case of complex-valued group velocities, which arise for the problem above. The existence and uniqueness theorems for these equations are proved and a complete picture of the. Whitham deformation is given for the case of a periodic one-gap boundary condition at the left infinity and zero at the right infinity.
Lipschitz regularity for certain problems from relaxation
Authors: Fuchs Martin
Published: November 2019

We prove local Lipschitz regularity for minimizers of functionals with integrand of polynomial growth near infinity. In contrast to Giaquinta and Modica (see [4]) we do not impose any differentiability or convexity conditions on the data.
Limit behavior of a shell, when first the thickness approaches zero and then the shell becomes a plate
Authors: Lods V.
Published: November 2019

We consider a shell, when first the thickness approaches zero and then the middle surface becomes flat. As noted by P.G. Ciarlet, the limit displacement obtained in this fashion is different from the limit displacement obtained when the shell becomes a plate first. Indeed, the ‘membrane’ and ‘bending’ effects both appear when first the shell becomes a plate, while only one of these effects appear when first the thickness approaches zero. This result is proved for shells with uniformly elliptic middle surfaces and for cylindrical shells.
Exponential attractors for a partially dissipative reaction system
Authors: Fabrie P. , Galusinski C.
Published: November 2019

After having established the existence of smooth absorbing sets, thanks to suitable a priori estimates, we obtain for a class of partially dissipative reaction systems a property known as squeezing property. This last leads to the existence of exponential attractors for which the fractal dimension is finite. Some examples of systems, borrowed from biology or chemistry, are given.
Asymptotics of resonances and eigenvalues for nonhomogeneous damped string
Authors: Shubov Marianna A.
Published: November 2019

We consider a semi-infinite and a finite string, both with a nonconstant density and subject to a positive viscous damping. In the first case it is assumed that both the density nonhomogeneity and the damping are concentrated on the finite interval [0,a]. Our main results are explicit asymptotic formulas for the resonances in the case of the semi-infinite string and for the eigenvalues in the case of the finite string. These formulas are obtained for three types of the density ρ(x): (i) ρ is bounded on [0,a] and (in the case of semi-infinite string) has a finite jump at x=a; (ii) ρ(a−0)=∞ (infinite condensation at x=a); (iii) ρ(a−0)=0 (infinite rarefaction at x=a). The result of this work is the first step in the spectral analysis of non-selfadjoint operator pencils generated by the equation of a nonhomogeneous damped string. Based on the results of this paper it will be shown in our forthcoming work that the systems of the quasimodes (of the infinite string) and the eigenmodes (of the finite string) have the Riesz-basis property.
Asymptotic analysis of kinetic equations for modeling a Schottky diode
Authors: Degonda P. , Poupaudb F. , Schmeiserc C. , Yamnahakkib A.
Published: November 2019

The aim of this work is to extend the mathematical model of a Schottky diode proposed in [5]. This model relies on an asymptotic analysis of the Vlasov–Poisson system. In [5], a one-dimensional model was derived. In the present paper, a three-dimensional model is investigated.
Anharmonic oscillator in high dimension
Authors: Sordoni Vania
Published: November 2019

In this paper we study the bottom of the spectrum of the semiclassical anharmonic oscillator Pm(h)=−h2Δm+Vm(x) where Vm(x)=μΣj=1mx2j+((g)/(mn−1))(Σj=1mxj2)n, μ∈R, g∈R+ and n∈N, n>1, when the number m of interacting particles is large. Denoting by λ(m,h) its lowest eigenvalue, we prove that lim m→+∞λ(m,h)/m exists and has a complete asymptotic expansion in powers of h, when the Planck's constant h tends to 0. For h fixed, we also obtain an expansion in powers of m−1 for the first eigenvalues of Pm. Moreover, we consider integrals of the form I(β,m)=∫Rm=e−βVm(x) dx where β is a large parameter and we prove the existence of the limit, as m→+∞, of the quantity (1/m)ln I(β,m) and that this limit has an asymptotic expansion in power of β−1 for large values of β.