Asymptotic Analysis - Volume 89, issue 1-2

Authors: Dai, Dan | Qiu, Weiyuan | Wang, Jun

Article Type: Research Article

Abstract: We consider the uniform asymptotics of polynomials orthogonal on [0,∞) with respect to the exponential weight w(x)=xα e−Q(x) , where α>−1 and Q(x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in different overlapping regions, respectively. These two regions together cover the whole complex plane. Our approach is based on the steepest descent method for Riemann–Hilbert problems introduced by Deift and Zhou [Ann. Math. 137 (1993), 295–368].

Keywords: orthogonal polynomial, uniform asymptotic expansion, Laguerre polynomial, Riemann–Hilbert problem

DOI: 10.3233/ASY-141219

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 1-19, 2014

Authors: Yeressian, Karen

Article Type: Research Article

Abstract: In this paper the asymptotic behavior of elliptic nonlocal equations in cylindrical domains is studied. These equations are associated with symmetric pure jump Lévy stochastic processes and are generalisation of equations involving the fractional Laplacian operator. First we consider the existence and uniqueness of weak solutions in appropriate Hilbert spaces associated with the Lévy measure and then study the asymptotic behavior in cylinders.

Keywords: asymptotic behavior, elliptic nonlocal equations, Lévy process

DOI: 10.3233/ASY-141224

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 21-35, 2014

Authors: Misiats, Oleksandr

Article Type: Research Article

Abstract: We study the minimization problem for simplified Ginzburg–Landau functional in doubly connected domain. This minimization problem is a subject to “semi-stiff” boundary conditions: |u|=1 and prescribed degrees p and q on the outer and inner boundaries respectively. Following the work of Berlyand and Rybalko [J. Eur. Math. Soc. 12 (2010), 1497–1531], we additionally prescribe the degree in the bulk (approximate bulk degree) to be d. The work [J. Eur. Math. Soc. 12 (2010), 1497–1531] established the sufficient conditions on the existence of Ginzburg–Landau minimizers, given in terms of p, q and d. The present work complements the result of [J. …Eur. Math. Soc. 12 (2010), 1497–1531] by providing the necessary conditions for the existence of nontrivial (nonconstant) minimizers. Show more

Keywords: Ginzburg–Landau minimizers, semi-stiff boundary conditions, approximate bulk degree

DOI: 10.3233/ASY-141227

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 37-61, 2014

Authors: Racca, Simone

Article Type: Research Article

Abstract: We study an evolution model for fractured elastic materials in the 2-dimensional case, for which the crack path is not assumed to be known a priori. We introduce some general assumptions on the structure of the fracture sets suitable to remove the restrictions on the regularity of the crack sets and to allow for kinking and branching to develop. In addition we define the front of the fracture and its velocity. By means of a time-discretization approach, we prove the existence of a continuous-time evolution that satisfies an energy inequality and a stability criterion. The energy balance also takes into …account the energy dissipated at the front of the fracture. The stability criterion is stated in the framework of Griffith's theory, in terms of the energy release rate, when the crack grows at least at one point of its front. Show more

Keywords: quasistatic crack evolution, dissipation, branching, kinking, Griffith's criterion

DOI: 10.3233/ASY-141233

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 63-110, 2014

Authors: Ene, Horia | Timofte, Claudia

Article Type: Research Article

Abstract: Using the periodic unfolding method, we shall analyze the asymptotic behavior of the solution of a system of coupled partial differential equations appearing in the modeling of an elasticity problem in a periodic structure formed by two interwoven and connected components with imperfect contact at the interface. Our setting proves to be relevant for analyzing contact problems for multiphase composites with imperfect interfaces.

Keywords: homogenization, linear elasticity, imperfect contact, the periodic unfolding method

DOI: 10.3233/ASY-141239

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 111-122, 2014

Authors: Hadiji, Rejeb | Yazidi, Habib

Article Type: Research Article

Abstract: We discuss the existence of solutions to the following nonlinear problem involving two critical Sobolev exponents −div (p(x)∇u)=β|u|2* −2 u+f(x,u)   in Ω, u≢0   in Ω, ∂u/∂ν=Q(x)|u|2* −2 u   on ∂Ω, where β≥0, Q is continuous on ∂Ω, p∈H1 (Ω) is continuous and positive in Ω¯ and f is a lower-order perturbation of |u|2* −1 with f(x,0)=0.

Keywords: Sobolev critical exponent, the trace embedding, variational problem, critical nonlinearity in the boundary, Palais–Smale condition, the mean curvature

DOI: 10.3233/ASY-141242

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 123-150, 2014

Authors: Figueiredo, Giovany M. | Santos, Jefferson A.

Article Type: Research Article

Abstract: This paper is concerned with the multiplicity of nontrivial solutions in an Orlicz–Sobolev space for a nonlocal problem with critical growth, involving N-functions and theory of locally Lipschitz continuous functionals. More precisely, in this paper, we study a result of multiplicity to the following multivalued elliptic problem: −M(∫Ω Φ(|∇u|) dx)ΔΦ u∈∂F(·,u)+αh(u)   in Ω, u∈W0 1 LΦ (Ω), where Ω⊂RN is a bounded smooth domain, N≥3, M is a continuous function, Φ is an N-function, h is an odd increasing homeomorphism from R to R, α is positive parameter, ΔΦ is the corresponding Φ-Laplacian and ∂F(·,t) …stands for Clarke generalized of a function F linked with critical growth. We use genus theory to obtain the main result. Show more

Keywords: Orlicz–Sobolev space, nonlocal problem, multivalued elliptic problem

DOI: 10.3233/ASY-141243

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 151-172, 2014

Authors: Sili, Ali

Article Type: Research Article

Abstract: We consider the stationary diffusion equation in a periodic composite medium made of two components Mε and Bε having very different diffusivity, the ratio between the coefficients of the diffusion in that structure being 1/αε 2 , where ε is the size of the period and αε which represents the amplitude of the diffusion in the inclusions Bε is a decreasing sequence towards zero. We show that the inclusions Bε work on the macroscopic diffusion as holes. In particular for scalings 0<αε �ε corresponding to very weak diffusion in Bε , the volume fraction of …the material at the limit is the well known one in the case of holes. Show more

Keywords: homogenization, high contrast, conductivity, degenerate equation, volume fraction

DOI: 10.3233/ASY-141241

Citation: Asymptotic Analysis, vol. 89, no. 1-2, pp. 173-187, 2014