Asymptotic Analysis - Volume 15, issue 3-4

Authors: Huang, Chaocheng

Article Type: Research Article

Abstract: This paper deals with the homogenization of the biharmonic equation Δ2 u=f in a domain containing randomly distributed tiny holes, with the Dirichlet boundary conditions. The size σ of the holes is assumed to be much smaller compared to the average distance ε between any two adjacent holes. We prove that as ε,σ→0, the solutions of the biharmonic equation converge to the solution of Δ2 u+κu=f, where κ depends on the shape of the holes and relative order of σ with respect to ε.

DOI: 10.3233/ASY-1997-153-401

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 203-227, 1997

Authors: Alexandre, Radjesvarane

Article Type: Research Article

Abstract: We give some results on homogenization of transport equations involving transverse and time-dependent oscillating coefficients. For this purpose, we adapt the results of Tartar, which enable to deduce the effective equations. These latter involve memory kernels, as expected, which are described in terms of pseudodifferential operators. Other types of partial differential equations are also dealt with here, which all involve first-order time derivatives.

Keywords: Homogenization, hyperbolic systems, linear differential equations, memory effects, neutronic theory, pseudodifferential operators, transport equations, transverse oscillations, Volterra equation

DOI: 10.3233/ASY-1997-153-402

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 229-259, 1997

Authors: Balser, Werner

Article Type: Research Article

Abstract: For general non-linear n-dimensional homogeneous systems of meromorphic ODE, we show existence of a complete formal solution, formally depending upon n free parameters. We investigate the structure of such formal solutions and, in particular, obtain a factorization, analogous to results of J.-P. Ramis and (independently) the author for linear systems.

DOI: 10.3233/ASY-1997-153-403

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 261-282, 1997

Authors: Hokari, Hisaaki | Matsumura, Akitaka

Article Type: Research Article

Abstract: This paper is concerned with the asymptotic behavior toward one-dimensional rarefaction wave for the solution of two-dimensional compressible Euler equation with an artificial viscosity. It is shown that if the initial data are suitably close to a constant state and their asymptotic values at x=±∞ are chosen so that the Riemann problem for the corresponding one-dimensional hyperbolic system admits the weak rarefaction wave, then the solution is proved to tend toward the one-dimensional rarefaction wave as t→+∞. The proof is given by using stability results of one-dimensional rarefaction wave and an elementary L2 energy method.

DOI: 10.3233/ASY-1997-153-404

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 283-298, 1997

Authors: Dal Maso, G. | Garroni, A.

Article Type: Research Article

Abstract: We prove that the asymptotic behaviour of the solutions of Dirichlet problems for second-order, linear, not necessarily symmetric elliptic equations in perforated domains of the form Ωh =Ω\Eh is uniquely determined by the asymptotic behaviour, as h→∞, of suitable capacities of the sets B∩Eh , where B runs in a conveniently large class of subsets of Ω.

DOI: 10.3233/ASY-1997-153-405

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 299-324, 1997

Authors: Bellettini, Giovanni | Colli Franzone, Piero | Paolini, Maurizio

Article Type: Research Article

Abstract: We study the convergence of the singularly perturbed anisotropic, nonhomogeneous reaction–diffusion equation ε∂t u−ε2 divT°(x,∇u)+f(u)−ε(c1 /c0 )g=0, where f is the derivative of a bistable quartic-like potential with unequal wells, T°(x,·) is a nonlinear monotone operator homogeneous of degree one and g is a given forcing term. More precisely, we prove that an appropriate level set of the solution satisfies an O(ε3 |log ε|2 ) error bound (in the Hausdorff distance) with respect to a hypersurface moving with the geometric law V=(c−εκϕ )nϕ +g-dependent terms, where nϕ is the so-called Cahn–Hoffmann vector and κϕ denotes the anisotropic mean curvature …of the hypersurface. We also discuss the connection between the anisotropic reaction–diffusion equation and the bidomain model, which is described by a system of equations modeling the propagation of an electric stimulus in the cardiac tissue. Show more

DOI: 10.3233/ASY-1997-153-406

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 325-358, 1997

Authors: Lesky, Peter H.

Article Type: Research Article

Abstract: We consider initial-boundary value problems for the plate equation u″(t,x)+Δ2 u(t,x)=f(x) in exterior domains of odd dimension. Especially the boundary condition Δu=∂Δu/∂n=0 is studied. We prove existence, uniqueness and time asymptotics for the solution. Furthermore, we study the plate equation with Dirichlet boundary condition. We give examples for right-hand sides f having unbounded support such that resonances occur.

DOI: 10.3233/ASY-1997-153-407

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 359-384, 1997

Authors: Truc, Françoise

Article Type: Research Article

Abstract: In this paper we investigate the asymptotic behaviour of the counting function of the eigenvalues for a semi-classical Schrödinger operator with a magnetic field, for a fixed energy, when the small parameter h goes to zero. We require for the magnetic field assumptions of the type “magnetic bottles” and we use a method of subdivision of Rd in cubes, in order to apply Courant's minimax variational principle. This method was previously used by Courant in the case of the classical counting function for minus Laplacian.

DOI: 10.3233/ASY-1997-153-408

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 385-395, 1997

Article Type: Other

Citation: Asymptotic Analysis, vol. 15, no. 3-4, pp. 397-398, 1997