Asymptotic Analysis - Volume 70, issue 1-2

Authors: Kwong, Man Kam | Wong, Solomon Wai-Him

Article Type: Research Article

Abstract: For many known examples of semilinear elliptic equations Δu+f(u)=0 in RN (N>1), a bounded radial solution u(r) converges to a constant as r→∞. Maier, in 1994, constructed, for N=2, an equation with a nonconvergent radial solution. Some necessary conditions for the existence of a nonconvergent solution were given by Maier, and later extended by Iaia. These conditions point out that, for N>2, equations with nonconvergent solutions are rather rare. A nonconvergent solution must oscillate between two constant values c1 <c2 and f must vanish at either c1 or c2 . In the neighborhood of one of …these points, f must fluctuate wildly in an unusual way that excludes almost all common functions. In this paper, we give a further improvement of the above result with an alternative, simpler proof. The proof depends on an elementary, but nonobvious property of an initial value problem. Show more

Keywords: radial solutions, semilinear elliptic equations

DOI: 10.3233/ASY-2010-0998

Citation: Asymptotic Analysis, vol. 70, no. 1-2, pp. 1-11, 2010

Authors: Bal, Guillaume | Pinaud, Olivier

Article Type: Research Article

Abstract: This paper analyzes the influence of general, small volume, inclusions on the trace at the domain's boundary of the solution to elliptic equations of the form ∇·Dε ∇uε =0 or (−Δ+qε )uε =0 with prescribed Neumann conditions. The theory is well known when the constitutive parameters in the elliptic equation assume the values of different and smooth functions in the background and inside the inclusions. We generalize the results to the case of arbitrary, and thus possibly rapid, fluctuations of the parameters inside the inclusion and obtain expansions of the trace of the solution at the domain's boundary up to …an order ε2d , where d is dimension and ε is the diameter of the inclusion. We construct inclusions whose leading influence is of order at most εd+1 rather than the expected εd . We also compare the expansions for the diffusion and Helmholtz equation and their relationship via the classical Liouville change of variables. Show more

Keywords: reconstruction of small volume inclusions, diffusion equation, Helmholtz equation, asymptotic expansions, inverse problems

DOI: 10.3233/ASY-2010-1002

Citation: Asymptotic Analysis, vol. 70, no. 1-2, pp. 13-50, 2010

Authors: Amaziane, B. | Pankratov, L. | Prytula, V.

Article Type: Research Article

Abstract: In this work we consider a model problem describing one phase flow through a thin porous layer made of weakly permeable porous blocks separated by thin fissures. The flow is modeled by a linear parabolic equation considered in a bounded 2D domain with high contrast coefficients. The problem involves three small parameters: the first one characterizes the periodicity of the distribution of the blocks in the layer, the second one stands for the thickness of the layer, the third one characterizes the volume fraction of the fissure part in the layer. Using the notion of two-scale convergence, we derive the …homogenized models which govern the global behavior of the flow when the small parameters tend to zero. The global models essentially depend on the relation between the small parameters. Show more

Keywords: double porosity models, fractured media, homogenization, thin layer, two-scale convergence

DOI: 10.3233/ASY-2010-1005

Citation: Asymptotic Analysis, vol. 70, no. 1-2, pp. 51-86, 2010

Authors: Giannoulis, Johannes | Mielke, Alexander | Sparber, Christof

Article Type: Research Article

Abstract: We investigate the asymptotic behavior of solutions to semi-classical Schrödinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras.

Keywords: nonlinear Schrödinger equation, Hartree nonlinearity, high-frequency asymptotics, WKB approximation

DOI: 10.3233/ASY-2010-1007

Citation: Asymptotic Analysis, vol. 70, no. 1-2, pp. 87-100, 2010

Authors: Freddi, Lorenzo | Paroni, Roberto | Zanini, Chiara

Article Type: Research Article

Abstract: A two-dimensional model which describes the evolution of a crack in a plate is deduced from a three-dimensional linearly elastic Griffith's type model. The result is achieved by adopting the framework of energetic solutions for rate-independent processes, to model three-dimensional fracture evolution, in conjunction with a variational dimension reduction procedure.

Keywords: dimension reduction, rate-independent processes, crack evolution, linear elasticity

DOI: 10.3233/ASY-2010-1003

Citation: Asymptotic Analysis, vol. 70, no. 1-2, pp. 101-123, 2010