Asymptotic Analysis - Volume 117, issue 3-4

Authors: Guo, Ting | Tang, Xianhua

Article Type: Research Article

Abstract: This paper is concerned with the following Choquard equation − Δ u + ( V ( x ) − μ | x | 2 ) u = ( ∫ R N F ( u ( y ) ) | x − y | α d y ) f ( u ) , u ∈ H 1 ( R N ) , …where N ⩾ 3 , 0 ⩽ μ < ( N − 2 ) 2 4 , 0 < α < N , V is 1-periodic in each of x 1 , x 2 , … , x N and F is the primitive function of f . Under some mild assumptions on V and f , we establish the existence and asymptotical behavior of ground state solutions by variational methods. Show more

Keywords: Choquard equation, ground state solutions, asymptotical behavior, inverse square potential

DOI: 10.3233/ASY-191549

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 141-160, 2020

Authors: Bárcena-Petisco, Jon Asier

Article Type: Research Article

Abstract: In this paper we consider a penalized Stokes equation defined in a regular domain Ω ⊂ R 2 and with Dirichlet boundary conditions. We prove that our system is null controllable using a scalar control defined in an open subset inside Ω and whose cost is bounded uniformly with respect to the parameter that converges to 0.

Keywords: Carleman inequality, penalized Stokes system, controllability

DOI: 10.3233/ASY-191550

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 161-198, 2020

Authors: Pruckner, Raphael

Article Type: Research Article

Abstract: We consider Jacobi matrices J whose parameters have the power asymptotics ρ n = n β 1 ( x 0 + x 1 n + O ( n − 1 − ϵ ) ) and q n = n β 2 ( y 0 + y 1 n + O ( n − 1 …− ϵ ) ) for the off-diagonal and diagonal, respectively. We show that for β 1 > β 2 , or β 1 = β 2 and 2 x 0 > | y 0 | , the matrix J is in the limit circle case and the convergence exponent of its spectrum is 1 / β 1 . Moreover, we obtain upper and lower bounds for the upper density of the spectrum. When the parameters of the matrix J have a power asymptotic with one more term, we characterise the occurrence of the limit circle case completely (including the exceptional case lim n → ∞ | q n | / ρ n = 2 ) and determine the convergence exponent in almost all cases. Show more

Keywords: Jacobi matrix, spectral analysis, difference equation, growth of entire function, canonical system, Berezanskiĭ’s theorem

DOI: 10.3233/ASY-191551

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 199-213, 2020

Authors: Charnley, M. | Vogelius, M.S.

Article Type: Research Article

Abstract: Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the conductivity of the inhomogeneity. The corresponding problem for thin inhomogeneities is not so well understood, in particular as concerns uniformity of the approximations. If the conductivity degenerates to 0 or goes to infinity as the width of the inhomogeneity goes to zero, the voltage potential may converge to different limiting solutions, and so the construction of uniform approximations is not straightforward. For the …case of thin two dimensional inhomogeneities with closed mid-curves such approximations were constructed and rigorously verified in (Chinese Annals of Mathematics, Series B 38 (2017 ) 293–344). The analysis relied heavily on the regularity of the approximate solutions. In this two part paper we continue this line of research, by showing that the same approximations remain valid, even when the mid-curve is open, and the corresponding approximate solutions have singularities at the endpoints of the curve. Show more

Keywords: Elliptic boundary-value problem, thin inhomogeneity, uniform approximation

DOI: 10.3233/ASY-191553

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 215-240, 2020

Authors: Charnley, M. | Vogelius, M.S.

Article Type: Research Article

Abstract: In this second part of a two part paper, we establish a local, uniform energy-approximation estimate for the solutions to a simplified model of thin inhomogeneities with open mid-curves. This local result plays a crucial role in the proof of the global, uniform approximation results established in the first part of this paper (Asymptotic Analysis (2019 )). For more details about the model we also refer the reader to (Chinese Annals of Mathematics, Series B 38 (2017 ) 293–344).

Keywords: Elliptic boundary-value problem, thin inhomogeneity, uniform approximation

DOI: 10.3233/ASY-191554

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 241-277, 2020

Authors: Pinar, Zehra

Article Type: Research Article

Abstract: The best known model of incompressible fluid is Boiti–Leon–Manna–Pempinelli (BLMP) equation. It is generally seen as two-dimensional model, also recently the model is seen in three dimensional and with variable coefficients. Till now, bilinear forms of the model are obtained. The exact solutions are not seen in the literature except special cases. This work investigates the exact solutions of the model with the strong methodologies based on auxiliary equation methods.

Keywords: Generalized Boiti–Leon–Manna–Pempinelli (BLMP) equation, Mathieu differential equation, exact solutions

DOI: 10.3233/ASY-191567

Citation: Asymptotic Analysis, vol. 117, no. 3-4, pp. 279-287, 2020