bluuebunny/crossref_metadata_2025 huggingface dataset:
Asymptotic controllability for linear hyperbolic systems
Authors: Li Tatsien , Rao Bopeng
Published: November 2019

For 1-D linear hyperbolic systems with constant coefficients we introduce the asymptotic controllability and the asymptotic zero controllability in L2 space under the lack of boundary controls and show the duality that they are equivalent, respectively, to the strong observability and the weak observability for the dual system. An example of 4×4 system with only one control is shown to be asymptotically controllable but not exactly controllable.
Spectral conditions for admissibility and observability of Schrödinger systems: Applications to finite element discretizations
Authors: Ervedoza Sylvain
Published: November 2019

In this article, we derive uniform admissibility and observability properties for the finite element space semi-discretizations of iż=A0z, where A0 is an unbounded self-adjoint positive definite operator with compact resolvent. In order to address this problem, we present several spectral criteria for admissibility and observability of such systems, which will be used to derive several results for space semi-discretizations of iż=A0z. Our approach provides very general results, which stand in any dimension and for any regular mesh (in the sense of finite elements). We also present applications to admissibility and observability for fully discrete approximation schemes, and to controllability and stabilization issues.
p-Laplacian problems with critical Sobolev exponent
Authors: Palatucci Giampiero
Published: November 2019

We use variational methods to study the asymptotic behavior of solutions of p-Laplacian problems with nearly subcritical non-linearity in general, possibly non-smooth, bounded domains.
A note on maximal solutions of nonlinear parabolic equations with absorption
Authors: Véron Laurent
Published: November 2019

If Ω is a bounded domain in RN and f a continuous nondecreasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): ∂tu−Δu+f(u)=0 in Q∞Ω:=Ω×(0,∞), u=∞ on the parabolic boundary ∂pQ. We prove that in most cases, the existence and uniqueness is reduced to the same property for the associated stationary equation in Ω.
The grazing collision limit of Kac caricature of Bose–Einstein particles
Authors: Allemand Thibaut , Toscani Giuseppe
Published: November 2019

We discuss the grazing collision limit of certain kinetic models of Bose–Einstein particles obtained from a suitable modification of the one-dimensional Kac caricature of a Maxwellian gas without cut-off. We recover in the limit a non-linear Fokker–Planck equation which presents many similarities with the one introduced by Kaniadakis and Quarati in Phys. Rev. E 49 (1994), 5103–5110. In order to do so, we perform a study of the moments of the solution. Moreover, as is typical in Maxwell models, we make an essential use of the Fourier version of the equation.
A lower semicontinuity result in SBD for surface integral functionals of Fracture Mechanics
Authors: Gargiulo Giuliano , Zappale Elvira
Published: November 2019

A lower semicontinuity result is proved in the space of special vector fields with bounded deformation for a fracture energetic model of the type

∫JuΨ([u], νu) dℋN−1,  [u]·νu≥0, ℋN−1 -a.e. on Ju.

A representation of the energy density Ψ, which ensures lower semicontinuity, is also given.
Stabilization of a piezoelectric system
Authors: Ammari Kaïs , Nicaise Serge
Published: November 2019

We consider a stabilization problem for a piezoelectric system. We prove an exponential stability result under some Lions geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate observability estimate.
On the Thomas–Fermi ground state in a harmonic potential
Authors: Gallo Clément , Pelinovsky Dmitry
Published: November 2019

We study non-linear ground states of the Gross–Pitaevskii equation in the space of one, two and three dimensions with a radially symmetric harmonic potential. The Thomas–Fermi approximation of ground states on various spatial scales was recently justified using variational methods. We justify here the Thomas–Fermi approximation on an uniform spatial scale using the Painlevé-II equation. In the space of one dimension, these results allow us to characterize the distribution of eigenvalues in the point spectrum of the Schrödinger operator associated with the non-linear ground state.
Limit cases in an elliptic problem with a parameter-dependent boundary condition
Authors: García-Melián Jorge , Rossi Julio D. , Sabina de Lis José C.
Published: November 2019

In this work we discuss existence, uniqueness and asymptotic profiles of positive solutions to the quasilinear problem

 −Δpu+a(x)up−1=−ur in Ω,

 |∇u|p−2 ∂u/∂ν=λup−1 on ∂Ω

for λ∈R, where r>p−1>0, a∈L∞(Ω). We analyze the existence of solutions in terms of a principal eigenvalue, and determine their asymptotic behavior both when r→p−1 and when r→∞.
Stochastic two-scale convergence of an integral functional
Authors: Sango Mamadou , Woukeng Jean Louis
Published: November 2019

In this paper we discuss the concept of stochastic two-scale convergence, which is appropriate to solve coupled-periodic and stochastic homogenization problems. This concept is a combination of both well-known two-scale convergence and stochastic two-scale convergence in the mean schemes, and is a generalization of the said previous methods. By way of illustration we apply it to solve a homogenization problem related to an integral functional with convex integrand. This problematic relies on the notion of dynamical system which is our basic tool.
Eigenvalue asymptotics for a star-graph damped vibrations problem
Authors: Pivovarchik Vyacheslav , Woracek Harald
Published: November 2019

We consider a boundary value problem generated by Sturm–Liouville equations on the edges of a star-shaped graph. Thereby a continuity condition and a condition depending on the spectral parameter is imposed at the interior vertex, corresponding to the case of damping in the problem of small transversal vibrations of a star graph of smooth inhomogeneous strings. At the pendant vertices Dirichlet boundary conditions are imposed. We describe the eigenvalue asymptotics of the problem under consideration.
Spectral properties of combinatorial Schrödinger operators on infinite weighted graphs
Authors: Akkouche Sofiane
Published: November 2019

Let H(λ)=−Δ+λb be the combinatorial Schrödinger operator on an infinite connected graph G with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)⊂[0,2]. When b≢0, let s(−Δ+λb):=inf σ(−Δ+λb) and M(−Δ+λb):=sup σ(−Δ+λb) be the bounds of the spectrum of the Schrödinger operator.

One of the aims of this paper is to study the influence of the potential b on the bounds of the spectrum of −Δ. More precisely, we give a condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar result for the top of the spectrum. We also prove that for a recurrent bipartite weighted graph, the only Schrödinger operator H=−Δ+b such that σ(H)⊂[0,2] is the Laplacian −Δ.

We study independently the asymptotic behaviour of the function λ↦s(−Δ+λb) for non-negative potentials b. We prove that s(∞):=lim λ→+∞s(−Δ+λb)<+∞ if and only if inf x∈Vb(x)=0 and we characterize this limit in the general case. As an application, we give an exact value of this limit for certain Jacobi matrices.
On a homogenization technique for singular integrals
Authors: Anza Hafsa Omar , Leghmizi Mohamed Lamine , Mandallena Jean-Philippe
Published: November 2019

We study homogenization by Γ-convergence of functionals of type

∫ΩW(x/ε,∇ϕ(x)) dx,

where Omega⊂RN is a bounded open set, ϕ∈W1,p(Omega;Rm) and p>1, when the 1-periodic integrand W :RN×Mm×N→[0,+∞] is not of p-polynomial growth. Our homogenization technique can be applied when m=N and W has a singular behavior of type W(x,ξ)→+∞ as det ξ→0. However, our technique is not consistent with the constraint W(x,ξ)=+∞ if and only if det ξ≤0.
Convergence estimates for abstract second-order singularly perturbed Cauchy problems with Lipschitzian nonlinearities
Authors: Perjan Andrei , Rusu Galina
Published: November 2019

We study the behavior of solutions to the problem

ε(u″ε(t)+A1uε(t))+u′ε(t)+A0uε(t)+B(uε(t))=fε(t), t∈(0,T],

uε(0)=u0ε, u′ε(0)=u1ε,

in the Hilbert space H as ε→0, where A1, A0 are two linear self-adjoint operators and B is a Lipschitzian operator.
The semi-classical Maupertuis–Jacobi correspondence for quasi-periodic Hamiltonian flows with applications to linear water waves theory
Authors: Dobrokhotov Sergey , Rouleux Michel
Published: November 2019

We extend to the semi-classical setting the Maupertuis–Jacobi correspondence for a pair of Hamiltonians (H(x,hDx),ℋ(x,hDx)). If ℋ(x,p) is completely integrable, or has merely an invariant Diophantine torus Λ in energy surface {ℋ=ℰ}, then we can construct a family of quasi-modes for H(x,hDx) at the corresponding energy E. This applies in particular to the linear theory of water waves, and determines trapped modes by an island, from the knowledge of Liouville metrics.
Global attractors for a vegetation model
Authors: Goto Yukie
Published: November 2019

In this article, a rigorous mathematical treatment of the dryland vegetation model introduced by Gilad et al. [Phys. Rev. Lett. 98(9) (2004), 098105-1–098105-4, J. Theoret. Biol. 244 (2007), 680–691] is presented. We prove the existence and uniqueness of solutions in (L1(Ω))3 and the existence of global attractors in L1(Ω;𝒟), where 𝒟 is an invariant region for the system. A key step is the regularization of the model by adding εΔ to the diffusion term and by approximating the initial data U0 by a sequence {U0,n} of smooth functions in (L1(Ω))3. The various a priori estimates and the maximum principle permit the passage to the limit as ε→0 and n→∞, proving the existence and uniqueness of solutions U in the specified space. Also, we deduce from the a priori estimates that the solution meets the necessary hypotheses (see Theorem 1.1 in Chapter 1 of Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, 1997) and hence, we obtain the existence of global attractors.
On the uniqueness of solution Cauchy's inverse problem for the equation Δu=au+b
Authors: Bezrodnykh S.I. , Demidov A.S.
Published: November 2019

The inverse Cauchy problem

Δu(x)=au(x)+b≥0  for x∈ω,  and  u=0,   ∂u/∂ν=Φ   on γ

is shown as having a unique solution within a wide class of simply connected domains ω⋐R2 with smooth boundary γ. Here a and b are real numbers to be determined, and Φ is a given function which is normalized by the condition ∫γΦ ds=1.
Corrections to Kirchhoff's law for the flow of viscous fluid in thin bifurcating channels and pipes
Authors: Lenzinger Michael
Published: November 2019

We consider the flow of a viscous Newtonian fluid in a bifurcation of thin pipes with a diameter-to-length ratio of order O(ε). The model is based on the stationary Navier–Stokes equations with pressure conditions on the outflow boundaries. Existence and local uniqueness is established under the assumption of small data represented by a Reynolds number Reε of order O(ε). We construct an asymptotic expansion in powers of ε and Reε for the solution consisting of Stokes flow in the junction part of the bifurcation and Poiseuille flow in the pipes. We introduce a correction to Kirchhoff's law of the balancing fluxes in O(ε) which allows to establish error estimates for the gradient of velocity. These estimates result from the analysis of the decay properties of the flow in the layer near the bifurcation.
Mechanical modeling of the skin
Authors: Blasselle A. , Griso G.
Published: November 2019

The skin is made of three main layers which are, from the top to the bottom: the epidermis, the dermis and the hypodermis. We consider the dermis as made of a Stokes fluid interacting with a periodic network of elastic fibers, assumed to obey the linearized elasticity law of behaviour. Above and below, the epidermis and the hypodermis are elastic solids. As the dimension of the thickness is very small compared to the two others, we assume periodic boundary conditions in those two planar directions. We study the 3d fluid–structure interaction system in a first part, and in a second part, we make the characteric size of the periodic element of the network go to zero in order to find an homogenized law for the whole skin. Starting from linear elastic materials, we find a viscoelastic law at the limit.
Resonance theory for perturbed Hill operator
Authors: Korotyaev Evgeny
Published: November 2019

We consider the Schrödinger operator Hy=−y″+(p+q)y with a periodic potential p plus a compactly supported potential q on the real line. The spectrum of H consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap γn≠∅,n≥1. We prove the following results: (1) the distribution of resonances in the disk with large radius is determined, (2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, (3) if H has infinitely many open gaps in the continuous spectrum, then for any sequence (κ)1∞,κn∈{0,2}, there exists a compactly supported potential q with ∫Rq dx=0 such that H has κn eigenvalues and 2−κn antibound states (resonances) in each gap γn for n large enough.
A remark on asymptotics of solutions to Schrödinger equation with fourth-order dispersion
Authors: Segata Jun-Ichi
Published: November 2019

We give an asymptotic formula in time of solutions to the linear fourth-order Schrödinger type equation in one space dimension without the mean zero condition on the initial data. Then we apply this formula to study the long time behavior of solution to the fourth-order Schrödinger type equation with the cubic non-linearity.
Smoothness of non-linear and non-separable subdivision schemes
Authors: Matei Basarab , Meignen Sylvain , Zakharova Anastasia
Published: November 2019

We study in this paper non-linear subdivision schemes in a multivariate setting allowing arbitrary dilation matrix. We investigate the convergence of such iterative process to some limit function. Our analysis is based on some conditions on the contractivity of the associated scheme for the differences. In particular, we show the regularity of the limit function, in Lp and Sobolev spaces.
Resonant states for a three-body problem under an external field
Authors: Grecchi Vincenzo , Kovařík Hynek , Martinez André , Sacchetti Andrea , Sordoni Vania
Published: November 2019

Here we consider one of the basic models for many-body problems under an external field: the molecule ion H2+ under the effect of an external Stark-type potential. If we consider the vibrational energy levels of the first two electronic states of the molecule ion H2+ then, in the semiclassical limit and by means of a suitable modified Born–Oppenheimer method, we can prove that they switch to sharp resonances localized in the same interval of energy of the vibrational levels when an external Stark-type field, with the same direction of the nuclear axis, occurs.
Asymptotic approximations for solutions to quasilinear and linear elliptic problems with different perturbed boundary conditions in perforated domains
Authors: Mel'nyk Taras A. , Sivak Olena A.
Published: November 2019

We consider quasilinear and linear boundary-value problems for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ωε that is ε-periodically perforated by small holes of order 𝒪(ε). The holes are divided into three ε-periodical sets depending on boundary conditions on their surfaces. On the boundaries of holes from one set we have the homogeneous Dirichlet conditions. On the boundaries of holes from the other sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For the solution to the quasilinear problem we find the leading terms of the asymptotics and prove the corresponding asymptotic estimates that show influence of the perturbation parameters. In the linear case we construct and justify the complete asymptotic expansion for the solution using two-scale asymptotic expansion method.
Rigorous derivation of the anelastic approximation to the Oberbeck–Boussinesq equations
Authors: Novotný Antonín , Růžička Michael , Thäter Gudrun
Published: November 2019

It is shown that the anelastic Oberbeck–Boussinesq system is a small Mach, small Péclet and small Froude number limit of the complete Navier–Stokes–Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows. This is a very useful information for numerical analysts.
A simplified model for elastic thin shells
Authors: Blanchard Dominique , Griso Georges
Published: November 2019

We introduce a simplified model for the minimization of the elastic energy in thin shells. This model is not obtained by an asymptotic analysis. The nonlinear simplified model admits always minimizers by contrast with the original one. We show the relevance of our approach by proving that the rescaled minimum of the simplified model and the rescaled infimum of the full model have the same limit as the thickness tends to 0. The simplified energy can be expressed as a functional acting over fields defined on the mid-surface of the shell and where the thickness remains as a parameter.
The effective Hamiltonian for thin layers with non-Hermitian Robin-type boundary conditions
Authors: Borisov Denis , Krejčiřík David
Published: November 2019

The Laplacian in an unbounded tubular neighbourhood of a hyperplane with non-Hermitian complex-symmetric Robin-type boundary conditions is investigated in the limit when the width of the neighbourhood diminishes. We show that the Laplacian converges in a norm resolvent sense to a self-adjoint Schrödinger operator in the hyperplane whose potential is expressed solely in terms of the boundary coupling function. As a consequence, we are able to explain some peculiar spectral properties of the non-Hermitian Laplacian by known results for Schrödinger operators.
Propagation of chaos for many-boson systems in one dimension with a point pair-interaction
Authors: Ammari Z. , Breteaux S.
Published: November 2019

We study the mean field limit of nonrelativistic quantum many-boson systems with delta potential in one-dimensional space. Such problem is related to the semiclassical limit of a second quantized Hamiltonian with an interaction given by a quartic Wick product. In this framework, we show that the evolution of coherent states is semiclassically given by squeezed coherent states under the action of a time-dependent affine Bogoliubov transformation. Results similar to those stated by Hepp [Comm. Math. Phys. 35 (1974), 265–277] and Ginibre-Velo [Comm. Math. Phys. 66 (1979), 37–76 and 68 (1979), 45–68] are proved. Furthermore, we show propagation of chaos for Schrödinger dynamics in the mean field limit using the argument of Rodnianski–Schlein [Comm. Math. Phys. 291 (2009), 31–61]. Thus, we provide a derivation of the cubic NLS equation in one dimension.
Nonclassical problems with nonlocal initial conditions for second-order evolution equations
Authors: Avalishvili G. , Avalishvili M. , Miara B.
Published: November 2019

In this paper we consider the nonclassical initial and initial-boundary value problems for second-order evolution equations with nonlocal initial conditions. We study the existence and uniqueness, in suitable abstract spaces, of solution of vector-valued distributions for such problems. We also present an algorithm of approximation of the solution of nonclassical problem with nonlocal initial conditions by a sequence of solutions of the corresponding classical problems. As an example we investigate the case of nonclassical hyperbolic equations and systems. In this particular case we show that the existence and uniqueness of solutions rely on algebraic properties of the ratios of values of the time variable in the nonlocal initial conditions and sizes of the spatial domain.
Global existence and regularity of the weakly compressible Navier–Stokes system
Authors: Jiang Ning , Levermore C. David
Published: November 2019

We construct the weakly nonlinear-dissipative approximate system for the general compressible Navier–Stokes system in a periodic domain. It was shown in Arch. Rational Mech. Anal. 201 (2011), 377–412, that because the Navier–Stokes system has an entropy structure, its approximate system will have Leray-like global weak solutions. These solutions decompose into an incompressible part governed by an incompressible Navier–Stokes system, and an acoustic part governed by a nonlocal quadratic equation which couples it to the incompressible part. We obtain regularity results for the acoustic part of the solution via a Littlewood–Paley decomposition that extend to this general setting results found by Masmoudi [Ann. Inst. H. Poincaré 18 (2001), 199–224] and Danchin [Amer. J. Math. 124 (2002), 1153–1219] in the γ-law barotropic setting.
Interface dynamics of the porous medium equation with a bistable reaction term
Authors: Alfaro Matthieu , Hilhorst Danielle
Published: November 2019

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.
On eigenfunctions corresponding to a small resurgent eigenvalue
Authors: Getmanenko Alexander
Published: November 2019

The paper is devoted to some foundational questions in resurgent analysis. As a main technical result, it is shown that under appropriate conditions the infinite sum of endlessly continuable majors commutes with the Laplace transform. A similar statement is proven for compatibility of a convolution and of an infinite sum of majors. We generalize the results of Candelpergher–Nosmas–Pham and prove a theorem about substitution of a small (extended) resurgent function into a holomorphic parameter of another resurgent function. Finally, we discuss an application of these results to the question of resurgence of eigenfunctions of a one-dimensional Schrödinger operator corresponding to a small resurgent eigenvalue.
Quantum revivals in two degrees of freedom integrable systems: The torus case
Authors: Lablée Olivier
Published: November 2019

The paper deals with the semi-classical behaviour of quantum dynamics for a semi-classical completely integrable system with two degrees of freedom near Liouville regular torus. The phenomenon of wave packet revivals is demonstrated in this article. The framework of this paper is semi-classical analysis (limit: h→0). For the proofs we use standard tools of real analysis, Fourier analysis and basic analytic number theory.
Compressible flows: New existence results and justification of the Reynolds asymptotic in thin films
Authors: Chupin Laurent , Sart Rémy
Published: November 2019

In this paper, we are first interested in the compressible Navier–Stokes equations with density dependent viscosities in bounded domains with non-homogeneous Dirichlet conditions. We study the well posedness of such models with density dependent viscosity coefficients in the non-stationary and in the stationary cases. We apply the last result in thin domains context, justifying the compressible Reynolds equations.
Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations
Authors: Mitake Hiroyoshi , Tran Hung V.
Published: November 2019

We investigate the large-time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton–Jacobi equations on the n-dimensional torus. We establish a convergence result to asymptotic solutions as time goes to infinity under rather restricted assumptions.
On the global existence for the axisymmetric Euler–Boussinesq system in critical Besov spaces
Authors: Sulaiman Samira
Published: November 2019

This paper is devoted to the global existence and uniqueness results for the three-dimensional Boussinesq system with axisymmetric initial data v0∈B5/22,1(R3) and ρ0∈B1/22,1(R3)∩Lp(R3) with p>6. This system couples the incompressible Euler equations with a transport-diffusion equation governing the density. In this case the Beale–Kato–Majda criterion (see [2]) is not known to be valid and to circumvent this difficulty we use in a crucial way some geometric properties of the vorticity.
Remarks on the ill-posedness of the Prandtl equation
Authors: Gérard-Varet D. , Nguyen T.
Published: November 2019

In the lines of the recent paper [J. Amer. Math. Soc. 23(2) (2010), 591–609], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C∞ initial data, local in time H1 solutions of the linearized Prandtl equation do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.
An a priori estimate for the singly periodic solutions of a semilinear equation
Authors: Allain Geneviève , Beaulieu Anne
Published: November 2019

We consider the positive solutions u of −Δu+u−up=0 in [0,2π]×RN−1, which are 2π-periodic in x1 and tend uniformly to 0 in the other variables. There exists a constant C such that any solution u verifies u(x1,x′)≤Cw0(x′) where w0 is the ground state solution of −Δv+v−vp=0 in RN−1. We prove that exactly the same estimate is true when the period is 2π/ε, even when ε tends to 0. We have a similar result for the gradient.
Eigenvalue bounds for radial magnetic bottles on the disk
Authors: Truc Françoise
Published: November 2019

We consider a Schrödinger operator HAD with a non-vanishing radial magnetic field B=dA and Dirichlet boundary conditions on the unit disk. We assume growth conditions on B near the boundary which guarantee in particular the compactness of the resolvent of this operator. Under some assumptions on an additional radial potential V the operator HAD−V has a discrete negative spectrum and we obtain an upper bound of the number of negative eigenvalues. As a consequence we get an upper bound of the number of eigenvalues of HAD smaller than any positive value λ, which involves the minimum of B and the square of the L2-norm of A(r)/r, where A(r) is the specific magnetic potential defined as the flux of the magnetic field through the disk of radius r centered in the origin.
An elliptic problem with two singularities
Authors: Croce Gisella
Published: November 2019

We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.
Boundary blow up solutions for fractional elliptic equations
Authors: Felmer Patricio , Quaas Alexander
Published: November 2019

In this article we study existence of boundary blow up solutions for some fractional elliptic equations including

(−Δ)αu+up=f in Ω,

u=g on Ωc,

lim x∈Ω,x→∂Ωu(x)=∞,

where Ω is a bounded domain of class C2, α∈(0,1) and the functions f :Ω→R and g :RN\Ω−→R are continuous. We obtain existence of a solution u when the boundary value g blows up at the boundary and we get explosion rate for u under an additional assumption on the rate of explosion of g. Our results are extended for an ample class of elliptic fractional nonlinear operators of Isaacs type.
Models for growth of heterogeneous sandpiles via Mosco convergence
Authors: Bocea M. , Mihăilescu M. , Pérez-Llanos M. , Rossi J.D.
Published: November 2019

In this paper we study the asymptotic behavior of several classes of power-law functionals involving variable exponents pn(·)→∞, via Mosco convergence. In the particular case pn(·)=np(·), we show that the sequence {Hn} of functionals Hn:L2(RN)→[0,+∞] given by

Hn(u)=∫RNλ(x)n/np(x)|∇u(x)|np(x) dx   if u∈L2(RN)∩W1,np(·)(RN),

+∞  otherwise,

converges in the sense of Mosco to a functional which vanishes on the set

u∈L2(RN): λ(x)|∇u|p(x)≤ 1 a.e. x∈RN

and is infinite in its complement. We also provide an example of a sequence of functionals whose Mosco limit cannot be described in terms of the characteristic function of a subset of L2(RN).

As an application of our results we obtain a model for the growth of a sandpile in which the allowed slope of the sand depends explicitly on the position in the sample.
Space–time semiclassical measures for three-dimensional elastodynamics: Boundary conditions for the hyperbolic set
Authors: Akian Jean-Luc
Published: November 2019

This paper is devoted to the linear elastodynamics equations in an open bounded set with a smooth boundary in the high-frequency limit. The boundary conditions are of Dirichlet or Neumann type. Semiclassical (or Wigner) measures enable to estimate the energy density related to these equations for high frequency phenomena. We determine here the boundary conditions verified by the space–time semiclassical measures related to the elastodynamics equations for the doubly hyperbolic set (under a non-interference hypothesis on the incident semiclassical measures) and for the hyperbolic–elliptic set. The non-interference hypothesis is similar to that of L. Miller in the case of transmission problems for the wave equation.
Large coupling limit for delta sphere interactions
Authors: BelHadjAli Hichem , BenAmor Ali
Published: November 2019

Let Hβ±:=−Δ±βδ(|x|=R), be the quantum mechanical Hamiltonian describing a delta interaction living on a sphere of radius R in three dimensions, with strength β>0. Let Dβ±:=(−Δ+1)−1−(Hβ±+1)−1 and D∞ be the strong limit w.r.t. β of the operators Dβ+. For suitable sequence (βn) we prove uniform convergence of Dβn− towards D∞ with optimal rate, namely βn−1. Convergence of the above mentioned operators against each other within Schatten–von Neumann ideals Sp, is established as well and the rate of convergence (depending on p) is determined. At the final step we discuss aspects of differences between negative and positive perturbations at large coupling.
Roughness effect on Neumann boundary condition
Authors: Chupin Laurent
Published: November 2019

We study the effect of a periodic roughness on a Neumann boundary condition. We show that, as in the case of a Dirichlet boundary condition, it is possible to approach this condition by a more complex law on a domain without rugosity, called wall law. This approach is however different from that usually used in Dirichlet case. In particular, we show that this wall law can be explicitly written using an energy developed in the roughness boundary layer. The first part deals with the case of a Laplace operator in a simple domain but many more general results are next given: when the domain or the operator are more complex or with Robin–Fourier boundary conditions. Some numerical illustrations are used to obtain magnitudes for the coefficients appearing in the new wall laws. Finally, these wall laws can be interpreted using a fictive boundary without rugosity. That allows to give an application to the water waves equation.
Semiclassical Szegö limit of resonance clusters for the hydrogen atom Stark Hamiltonian
Authors: Hislop Peter D. , Villegas-Blas Carlos
Published: November 2019

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szegö-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szegö-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non-self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues.
Approximating travelling waves by equilibria of non-local equations
Authors: Arrieta Jose M. , López-Fernández María , Zuazua Enrique
Published: November 2019

We consider an evolution equation of parabolic type in R having a travelling wave solution. We study the effects on the dynamics of an appropriate change of variables which transforms the equation into a non-local evolution one having a travelling wave solution with zero speed of propagation with exactly the same profile as the original one. This procedure allows us to compute simultaneously the travelling wave profile and its propagation speed avoiding moving meshes, as we illustrate with several numerical examples. We analyze the relation of the new equation with the original one in the entire real line. We also analyze the behavior of the non-local problem in a bounded interval with appropriate boundary conditions. We show that it has a unique stationary solution which approaches the traveling wave as the interval gets larger and larger and that is asymptotically stable for large enough intervals.
Large liquidity expansion of super-hedging costs
Authors: Possamaï Dylan , Soner H. Mete , Touzi Nizar
Published: November 2019

We consider a financial market with liquidity cost as in [Çetin, Jarrow and Protter, Finance and Stochastics 8 (2004), 311–341], where the supply function Sε(s,ν) depends on a parameter ε≥0 with S0(s,ν)=s corresponding to the perfect liquid situation. Using the PDE characterization of Çetin, Soner and Touzi [Finance and Stochastics 14(3) (2010), 317–341], of the super-hedging cost of an option written on such a stock, we provide a Taylor expansion of the super-hedging cost in powers of ε. In particular, we explicitly compute the first term in the expansion for a European Call option and give bounds for the order of the expansion for a European Digital Option.
On metastability in nearly-elastic systems
Authors: Hu Wenqing
Published: November 2019

We consider a nearly-elastic model system with one degree of freedom. In each collision with the “wall”, the system can either lose or gain a small amount of energy due to stochastic perturbation. The weak limit of the corresponding slow motion, which is a stochastic process on a graph, is calculated. A large deviation type asymptotics and the metastability of the system are also considered.
Long time asymptotic expansions of heat kernels for perturbed Hill operators
Authors: Murata Minoru , Tsuchida Tetsuo
Published: November 2019

We obtain, by exploiting Jost functions, asymptotic expansions of Green functions of perturbed Hill operators with respect to the spectral parameter around the minima of the continuous spectra. From the Green function expansions, we derive asymptotic expansions for large time of heat kernels for perturbed Hill operators.
Permeability of interfaces with alternating pores in parabolic problems
Authors: Andreucci Daniele , Bellaveglia Dario
Published: November 2019

We study a parabolic problem set in a domain divided by a perforated interface. The pores alternate between an open and a closed state, periodically in time. We consider the asymptotics of the solution for vanishingly small size of the pores and time period. The interface condition prevailing in the limit is a linear relation between the flux (on either side) and the jump of the limiting solution across the interface. More exactly this behaviour only takes place when the relative sizes of the relevant geometrical and temporal parameters are connected by suitable relations.

With respect to the stationary version of this problem, which is known in the literature, we demonstrate the appearance of a new admissible asymptotic standard. More in general, we describe the precise interplay between the geometrical and temporal parameters leading to the quoted interface condition.

This work represents a preliminary mathematical investigation of a model of selective transport of chemical species through biological membranes.
Homogenization of a Stokes problem in a porous medium by the periodic unfolding method
Authors: Zaki Rachad
Published: November 2019

We consider the Stokes problem in a domain with holes periodically distributed with a period ε. The size of the holes is of the order of ε, a small parameter going to zero. On the boundary of the holes we prescribe a Robin-type condition depending on a parameter γ. The aim is to give the asymptotic behavior of the velocity and of the pressure of the fluid as ε goes to zero. The study for a problem of this type was done in Math. Meth. Appl. Sci. 19 (1996), 857–881, via classical homogenization methods. In this work we use the periodic unfolding method in perforated domains (see C. R. Acad. Sci. Paris, Série 1 342 (2006), 469–474; Portugaliae Mathematica 63(4) (2006), 467–496; Asymptotic Analysis 53(4) (2007), 209–235; in: Multiple Scales in Problems in Biomath., Mech., Physics and Numeric, Gakuto Int. Series, Math. Sci. App., Vol. 31, Gakkokotosho, 2009, pp. 37–68, and SIAM J. Math. Anal. 44(2) (2012), 718–760), which allows us to consider a general geometric framework. We give the limit problems corresponding to different values of γ (Darcy, Brinkmann or Stokes type problems).
Convergence of a sequence of solutions of the stochastic two-dimensional equations of second grade fluids
Authors: Razafimandimby Paul André , Sango Mamadou
Published: November 2019

We study the limit of the stochastic model for two dimensional second grade fluids subjected to the periodic boundary conditions as the stress modulus tends to zero. We show that under suitable conditions on the data the whole sequence of strong probabilistic solutions (uα) of the stochastic second grade fluid converges to the unique strong probabilistic solution of the stochastic Navier–Stokes equations.
Orthogonal trigonometric polynomials: Riemann–Hilbert analysis and relations with OPUC
Authors: Du Zhihua , Du Jinyuan
Published: November 2019

In this paper, we study the theory of orthogonal trigonometric polynomials (OTPs). We obtain asymptotics of OTPs with positive and analytic weight functions by Riemann–Hilbert approach and find that they have relations with orthogonal polynomials on the unit circle (OPUC). By the relations and the theory of OPUC, we also get four-terms recurrent formulae, Christoffel–Darboux formula and some algebraic and asymptotic properties of zeros for orthogonal trigonometric polynomials.
Existence and bounds for nonlinear Schrödinger systems with strong competition
Authors: Domingos Ana Rute , Ramos Miguel
Published: November 2019

We consider systems of coupled Schrödinger equations which appear in nonlinear optics and binary Bose–Einstein condensation. Namely, we prove that for most μ,ν∈R, a,b>0, the system −Δu=μu+au3−βuv2, −Δv=νv+bv3−βvu2, u,v∈H10(B1(0)), where B1(0) is the unit ball of R3, admits a family of radially symmetric positive solutions (uβ,vβ) provided the interaction parameter β>0 is sufficiently large. By using a Morse index technique we deduce that these solutions are bounded uniformly in β, hence their limit functions as β→∞ undergo the phenomenon of phase segregation.
Uniqueness and non-uniqueness of solutions to quasilinear parabolic equations with a singular coefficient on weighted Riemannian manifolds
Authors: Punzo Fabio
Published: November 2019

We study, on weighted Riemannian model manifolds, well posedness of the Cauchy problem for a class of quasilinear parabolic equations with a coefficient which can be singular at infinity. We establish uniqueness or non-uniqueness of bounded solutions, under suitable assumptions on the behavior at infinity of the singular coefficient and on the Green function for the weighted Laplace–Beltrami operator.
Bounded solutions to the 1-Laplacian equation with a critical gradient term
Authors: Andreu Fuensanta , Dall'Aglio Andrea , Segura de León Sergio
Published: November 2019

In this paper, we study the Dirichlet problem for an elliptic equation, in which the 1-Laplacian operator and lower-order terms appear. We introduce a suitable definition of solution and prove the existence of, at least, one bounded solution in BV(Ω) having a negligible jump part. Moreover, a uniqueness result for small positive data is proved, and explicit examples of solutions are shown.
On polynomial and exponential decay of eigen-solutions to exterior boundary value problems for the generalized time-harmonic Maxwell system
Authors: Pauly Dirk
Published: November 2019

We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain Ω⊂RN, N≥1, with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a rate r−τ, τ>1, towards the identity. As a canonical application we show that the corresponding eigen-values do not accumulate and that by means of Eidus' limiting absorption principle a Fredholm alternative holds true.
Effective viscosity properties of dilute suspensions of arbitrarily shaped particles
Authors: Ammari Habib , Garapon Pierre , Kang Hyeonbae , Lee Hyundae
Published: November 2019

In this paper we derive high-order asymptotic expansions of the effective viscosity properties of a dilute periodic suspension composed of freely-suspended arbitrarily shaped particles dispersed in an incompressible Newtonian fluid. High-order terms are not only function of the viscous moment tensor but also of a distortion tensor that characterizes the periodic array.
A mixing effect induced by sources concentrated in a soft junction and the gradient concentration phenomenon
Authors: Bessoud Anne-Laure , Juntharee Pongpol , Licht Christian , Michaille Gérard
Published: November 2019

We show that the variational limit of a ε-soft and thin junction problem (𝒫ε) with sources concentrated in the junction gives rise to a surface energy mixing the internal energy and sources. The surface energy functional possesses an integral representation with respect to the gradient Young-concentration measures generated by sequences (ūε)ε>0 of minimizers of (𝒫ε).
Discrete spectrum of quantum Hall effect Hamiltonians II: Periodic edge potentials
Authors: Miranda Pablo , Raikov Georgi
Published: November 2019

We consider the unperturbed operator H0:=(−i∇−A)2+W, self-adjoint in L2(R2). Here A is a magnetic potential which generates a constant magnetic field b>0, and the edge potential W=W¯ is a 𝒯-periodic non-constant bounded function depending only on the first coordinate x∈R of (x,y)∈R2. Then the spectrum σ(H0) of H0 has a band structure, the band functions are b𝒯-periodic, and generically there are infinitely many open gaps in σ(H0). We establish explicit sufficient conditions which guarantee that a given band of σ(H0) has a positive length, and all the extremal points of the corresponding band function are non-degenerate. Under these assumptions we consider the perturbed operators H±=H0±V where the electric potential V∈L∞(R2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H± in the spectral gaps of H0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schrödinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum σ(H0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.
Homogenization results for elliptic problems in periodically perforated domains with mixed-type boundary conditions
Authors: Capatina Anca , Ene Horia , Timofte Claudia
Published: November 2019

The asymptotic behaviour of a class of elliptic equations with highly oscillating coefficients, in a perforated periodic domain, is analyzed. We consider, in each period, two types of holes and we impose, on their boundaries, a Signorini and, respectively, a Neumann condition. Using the periodic unfolding method, we prove that the limit problem contains two additional terms, a right-hand side term and a “strange” one.
Asymptotic expansion of the scattering matrix associated with matrix Schrödinger operator
Authors: Baklouti Hamadi , Abdeljeilil Slaheddine Ben
Published: November 2019

In this paper, we study the scattering theory for a 2×2 matrix Schrödinger operator

P=−h2d2/dx2I2+V(x)+hR(x,hDx)

on L2(R)⌖L2(R), where V(x) is a real diagonal matrix, the eigenvalues of which are never equal. Under some assumptions of analyticity and decay at infinity of V, we describe the asymptotic behavior of the scattering matrix S=(sij)1≤i,j≤4 associated with P when the semi-classical parameter h goes to zero. Moreover, we obtain the estimate

‖S12‖+‖S21‖=O(e−δ/h),

where S12 and S21 are the two off-diagonal elements of S and δ>0 is a constant which is explicitly related to the behavior of V(x) in the complex domain.
Uniform attractor for non-autonomous extensible beam equation
Authors: Kang Jum-Ran
Published: November 2019

In this paper, we prove the existence of a uniform attractor for non-autonomous extensible beam equation with localized damping.
Stabilization of unbounded linear control systems in Hilbert space with monotone feedback
Authors: Berrahmoune Larbi
Published: November 2019

We consider linear control systems of the form y′(t)=Ay(t)+Bu(t) where A generates a strongly continuous semigroup of contractions (etA)t≥0 on an infinite-dimensional Hilbert space Y. We suppose that the control is unbounded in the sense that the linear control operator B is bounded from the (Hilbert) control space U to some larger space W such that Y⊂W, but not from U into Y. Taking into account eventual control saturation, we study the problem of stabilization by (possibly nonlinear) monotone feedback of the form u(t)=−f(B*y(t)). We extend to the unbounded monotone feedback context weak and strong stability results which have been established for linear feedback systems. These results are based on weak observability involving the unstable subspace. This fact is illustrated by a heat equation with singular control. In the particular case where the system can be reduced to a second-order evolution equation of the form z″(t)+𝒜z(t)=ℬu(t), we establish decay estimates with (eventually) saturating feedback and under a strong observability property. We show that when the initial data are sufficiently regular, we obtain exponential and polynomial decay. We establish also that when the control operator ℬ is bounded, the regularity assumption can be dispensed with. Applications to the wave equation with distributed control and pointwise control are considered. We establish for these systems non-uniform exponential decay with general saturating feedback. For a string, we present an explicit saturating feedback leading to exponential decay of the energy. Moreover, the degree of regularity of the initial data are related to the Diophantine properties of the actuator position.
Homogenization of rate-dependent inelastic models of monotone type
Authors: Nesenenko Sergiy
Published: November 2019

Using the periodic unfolding method we derive the homogenized equations for the quasi-static initial boundary value problem with internal variables, which model the deformation behavior of viscoplastic materials with a periodic microstructure. The free energy associated with the problem is allowed to be positive semi-definite.