Asymptotic Analysis - Volume 31, issue 3-4

Authors: Lods, Véronique | Mardare, Cristinel

Article Type: Research Article

Abstract: We consider a shell, i.e., a three‐dimensional body with a small thickness (denoted by 2ε), which is clamped along its entire lateral boundary and subjected to regular loads. In the linear case, one can use the two‐dimensional models of Koiter or Naghdi to calculate the displacement vector field of the shell. Some error estimates have been obtained in [12,13] between these models and the three‐dimensional displacement for flexural or membrane shells. Here, we do not make any assumptions on the geometry of the shell. In particular, the space of inextensional displacements can be reduced to zero. The assumptions on the …loads are weak: for the sake of simplicity, one can consider regular loads (in H1 ) which do not depend on the transverse variable. We then establish a relative error estimate for the scaled linearized deformation tensor between Koiter's model (or Naghdi's model) and the three‐dimensional model. These estimates hold though the limit model is not always known. In addition, further assumptions on the data allow to recover the error estimates concerning the displacements which have been proved in [12,13]. Show more

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 189-210, 2002

Authors: Schmidt, D. | Sibuya, Y. | Tabara, T.J.

Article Type: Research Article

Abstract: We have demonstrated previously how to calculate a Stokes multiplier concerning subdominant solutions of a second order differential equation with polynomial coefficients without utilizing any suitable integral representation of its solutions. With a recent discovery of a transformation, we can derive a new Stokes multiplier from the previously calculated multiplier.

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 211-227, 2002

Authors: Benachour, S. | Laurençot, Ph. | Schmitt, D. | Souplet, Ph.

Article Type: Research Article

Abstract: Extinction in finite time and non‐compactness of the support are investigated for non‐negative classical solutions to the Cauchy problem ut −Δu+|∇u|p =0 when p∈(0,1). The occurrence of these phenomena is shown to depend on the behaviour of the initial data u0 for large values of x. In particular, we obtain the optimal decay rate of u0 at infinity to ensure finite‐time extinction of u. The case of periodic boundary conditions is also considered. Finally the optimality of temporal L∞ ‐decay estimates obtained previously is discussed.

Keywords: extinction in finite time, non‐extinction, temporal decay estimates, viscous Hamilton–Jacobi equations

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 229-246, 2002

Authors: Mossino, Jacqueline | Vanninathan, Muthusamy

Article Type: Research Article

Abstract: This work considers a reinforcement of a multiconnected domain representing the cross section of a cylindrical bar. The model studied in this paper is motivated from the classical problem of elastic torsion of such a structure. The stress potential is assumed to satisfy an equation involving p‐Laplacian in the interior. Dirichlet boundary condition is imposed on the outer boundary. We seek the potential in the space of functions with trace being an unknown constant on the inner boundary. Consequently, there is an induced natural condition involving the integral of its conormal derivative on the inner boundary. We assume that the …reinforcement is nonhomogeneous, periodically oscillating and has small thickness which is of the same order as the period. We describe its limiting behaviour as the period goes to zero. The limit problem has certain new features which are pointed out. Show more

Keywords: reinforcement, boundary layer, Γ‐convergence, torsional rigidity, quasilinear elliptic problem

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 247-263, 2002

Authors: Hitrik, Michael

Article Type: Research Article

Abstract: The eigenfrequencies associated to a damped wave equation are known to belong to a band parallel to the real axis. Under the assumption of periodicity of the geodesic flow we study the asymptotic distribution of the eigenfrequencies in the band. We show that the set of eigenfrequencies exhibits a cluster structure determined by the Morse index of the closed geodesics and the damping coefficient averaged along the geodesic flow. The asymptotics for the multiplicities of the clusters are also obtained.

Keywords: damped wave equations, eigenvalue asymptotics, Zoll manifolds

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 265-277, 2002

Authors: Pladdy, Christopher

Article Type: Research Article

Abstract: We consider the Dirac operator with a scalar short‐range potential. Estimates for the extended resolvents, R± (λ), of this operator considered as a bounded operator from ℒs 2 into the weighted Sobolev space ℋ−s 1 are discussed. The extended resolvents R0 ± (λ) of the free Dirac operator are shown to be O(λ) as |λ|→∞ in the uniform operator topology of B(ℒs 2 ,ℋ−s 1 ), and this is shown to be the optimum result. R± (λ) are shown to be R0 ± (λ)+O(1) as |λ|→∞ in B(ℒs 2 ,ℋ−s 1 ). R± (λ)−R0 ± (λ) are shown …to converge strongly to zero in B(ℒs 2 ,ℋ−s 1 ). The method of proof uses results for the free Schrödinger operator resolvent and the free Dirac operator resolvent, both considered as operators from ℒs 2 to ℋ−s 1 , and the estimates for the free Dirac and Schrödinger resolvents, both considered as operators from ℒs 2 to ℒ−s 2 . Show more

Keywords: Dirac operators, limiting absorption principle, resolvent estimates, Sobolev space, pseudodifferential operator

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 279-295, 2002

Authors: Esposito, Antonio Corbo | D'Apice, Ciro | Gaudiello, Antonio

Article Type: Research Article

Abstract: We present an interference phenomenon in the homogenization of the Poisson equation when nonhomogeneous Neumann condition is imposed on part of the boundary of the holes and Dirichlet condition is imposed on the remaining part.

Keywords: homogenization, perforated domains

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 297-316, 2002

Authors: Faou, Erwan

Article Type: Research Article

Abstract: The three‐dimensional equations of elasticity are posed on a domain of $\mathbb{R}$ 3 defining a thin shell of thickness 2ε. The traction free conditions are imposed on the upper and lower faces together with the clamped boundary conditions on the lateral boundary. After a scaling in the transverse variable, the elasticity operator admits a power series expansion in ε with intrinsic coefficients with respect to the mean surface of the shell. This leads to define a formal series problem in ε associated with the three‐dimensional equations. The main result is the reduction of this problem to a formal …series boundary value problem posed on the mean surface of the shell. Show more

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 317-361, 2002

Article Type: Other

Citation: Asymptotic Analysis, vol. 31, no. 3-4, pp. 363-364, 2002