Asymptotic Analysis - Volume Pre-press, issue Pre-press

Authors: Hovsepyan, Narek

Article Type: Research Article

Abstract: We analyze an approximate interior transmission eigenvalue problem in R d for d = 2 or d = 3 , motivated by the transmission problem of a transformation optics-based cloaking scheme and obtained by replacing the refractive index with its first order approximation, which is an unbounded function. Using the radial symmetry we show the existence of (infinitely many) complex transmission eigenvalues and prove their discreteness. Moreover, it is shown that there exists a horizontal strip in the complex plane around the real axis, that does not contain any …transmission eigenvalues. Show more

Keywords: Born approximation, transmission eigenvalues, eigenvalue-free regions, inverse scattering, spherically stratified medium

DOI: 10.3233/ASY-231868

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-34, 2023

Authors: Zhang, Wen | Wu, Changxing | Ruan, Zhousheng | Qiu, Shufang

Article Type: Research Article

Abstract: In this article, we construct a Jacobi spectral collocation scheme to approximate the Caputo fractional derivative based on Jacobi–Gauss quadrature. The convergence analysis is provided in anisotropic Jacobi-weighted Sobolev spaces. Furthermore, the convergence rate is presented for solving Caputo fractional derivative with noisy data by invoking the mollification regularization method. Lastly, numerical examples illustrate the effectiveness and stability of the proposed method.

Keywords: Fractional derivative, Jacobi collocation, mollification, Gaussian quadrature

DOI: 10.3233/ASY-231869

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-17, 2023

Authors: Mel’nyk, Taras | Rohde, Christian

Article Type: Research Article

Abstract: This article completes the study of the influence of the intensity parameter α in the boundary condition ε ∂ ν ε u ε − u ε V ε → · ν ε = ε α φ ε given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order O ( ε ) . Inside …of the thin network a time-dependent convection-diffusion equation with high Péclet number of order O ( ε − 1 ) is considered. The novelty of this article is the case of α < 1 , which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity φ ε (the cases α = 1 and α > 1 were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution u ε as ε → 0 , i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter ε . Show more

Keywords: Asymptotic expansion, convection-diffusion problem, boundary interactions, thin graph-like junction, hyperbolic limit model

DOI: 10.3233/ASY-231876

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-26, 2023

Authors: Giacomoni, Jacques | Il’yasov, Yavdat | Kumar, Deepak

Article Type: Research Article

Abstract: We discuss the existence and non-existence of periodic in one variable and compactly supported in the other variables least energy solutions for equations with non-Lipschitz nonlinearity of the form: − Δ u = λ u p − u q in R N + 1 , where 0 < q < p < 1 and λ ∈ R . The approach is based on the Nehari manifold method supplemented by a one-sided constraint given through the functional of the suitable Pohozaev identity. …The limit value of the parameter λ , where the approach is applicable, corresponds to the existence of periodic in one variable and compactly supported in the other variables least energy solutions. This value is found through the extrem values of nonlinear generalized Rayleigh quotients and the so-called curve of the critical exponents of p , q . Important properties of the solutions are derived for suitable ranges of the parameters, such as that they are not trivial with respect to the periodic variable and do not coincide with compactly supported solutions on the entire space R N + 1 . Show more

Keywords: Semilinear elliptic equation, non-Lipschitz nonlinearity, compactly supported solutions, periodic solutions, generalized Rayleigh’s quotients, the Pohozaev identity

DOI: 10.3233/ASY-231878

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-25, 2023

Authors: Dai, Guowei | Zhang, Zhitao

Article Type: Research Article

Abstract: By bifurcation and topological methods, we study the existence/nonexistence and multiplicity of one-sign or nodal solutions of the following k -th mean curvature problem in Minkowski spacetime r N − k v ′ 1 − v ′ 2 k ′ = λ N C N k r N − 1 H k ( r , v ) …in  ( 0 , R ) , | v ′ | < 1 in  ( 0 , R ) , v ′ ( 0 ) = v ( R ) = 0 . As a previous step, we investigate the spectral structure of its linearized problem at zero. Moreover, we also obtain a priori bounds and the asymptotic behaviors of solutions with respect to λ . Show more

Keywords: Spectrum, Bifurcation, k-th mean curvature, Nodal solutions, A priori bounds, Asymptotic behaviors, One-sign solution

DOI: 10.3233/ASY-231877

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-33, 2023

Authors: Ayouch, C. | Meskine, D. | Tilioua, M.

Article Type: Research Article

Abstract: In this paper, the Landau–Lifshitz–Baryakhtar (LLBar) equation for magnetization dynamics in ferrimagnets is considered. We prove global existence of a periodic solutions as well as local existence and uniqueness of regular solutions. We also study the relationships between the Landau–Lifshitz–Baryakhtar equation and both Landau–Lifshitz–Bloch and harmonic map equations.

Keywords: Landau–Lifshitz–Baryakhtar equation, harmonic map equation, local well-posedness, asymptotic behaviour

DOI: 10.3233/ASY-231874

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-27, 2023

Authors: Shapiro, Jacob

Article Type: Research Article

Abstract: We consider, for h , E > 0 , resolvent estimates for the semiclassical Schrödinger operator − h 2 Δ + V − E . Near infinity, the potential takes the form V = V L + V S , where V L is a long range potential which is Lipschitz with respect to the radial variable, while V S = O ( | x | − 1 ( log | x | …) − ρ ) for some ρ > 1 . Near the origin, | V | may behave like | x | − β , provided 0 ⩽ β < 2 ( 3 − 1 ) . We find that, for any ρ ˜ > 1 , there are C , h 0 > 0 such that we have a resolvent bound of the form exp ( C h − 2 ( log ( h − 1 ) ) 1 + ρ ˜ ) for all h ∈ ( 0 , h 0 ] . The h -dependence of the bound improves if V S decays at a faster rate toward infinity. Show more

Keywords: Resolvent estimate, Schrödinger operator, short range potential

DOI: 10.3233/ASY-231872

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-24, 2023

Authors: Dechicha, Dahmane | Puel, Marjolaine

Article Type: Research Article

Abstract: In this paper, we extend the spectral method developed (Dechicha and Puel (2023 )) to any dimension d ⩾ 1 , in order to construct an eigen-solution for the Fokker–Planck operator with heavy tail equilibria, of the form ( 1 + | v | 2 ) − β 2 , in the range β ∈ ] d , d + 4 [ . The method developed in dimension 1 was inspired by the work of H. Koch on nonlinear KdV equation (Nonlinearity 28 …(2015 ) 545). The strategy in this paper is the same as in dimension 1 but the tools are different, since dimension 1 was based on ODE methods. As a direct consequence of our construction, we obtain the fractional diffusion limit for the kinetic Fokker–Planck equation, for the correct density ρ : = ∫ R d f d v , with a fractional Laplacian κ ( − Δ ) β − d + 2 6 and a positive diffusion coefficient κ . Show more

Keywords: Kinetic Fokker–Planck equation, Fokker–Planck operator, heavy-tailed equilibrium, anomalous diffusion, fractional diffusion, spectral theory, eigen-solutions

DOI: 10.3233/ASY-231870

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-54, 2023

Authors: Garg, Swati | Sardar, Bidhan Chandra

Article Type: Research Article

Abstract: The present article deals with the homogenization of a distributive optimal control problem (OCP) subjected to the more generalized stationary Stokes equation involving unidirectional oscillating coefficients posed in a two-dimensional oscillating domain. The cost functional considered is of the Dirichlet type involving a unidirectional oscillating coefficient matrix. We characterize the optimal control and study the homogenization of this OCP with the aid of the unfolding operator. Due to the presence of oscillating matrices both in the governing Stokes equations and the cost functional, one obtains the limit OCP involving a perturbed tensor in the convergence analysis.

Keywords: Homogenization, optimal control, oscillating boundary, unfolding operator, Stokes equations

DOI: 10.3233/ASY-231867

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-26, 2023

Authors: Jleli, Mohamed | Samet, Bessem | Vetro, Calogero

Article Type: Research Article

Abstract: We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions.

Keywords: Higher order evolution inequalities, Leray–Hardy potential, half ball, nonexistence result

DOI: 10.3233/ASY-231873

Citation: Asymptotic Analysis, vol. Pre-press, no. Pre-press, pp. 1-22, 2023