Asymptotic Analysis - Volume 9, issue 2

Authors: Bardos, Claude | Rauch, Jeffrey

Article Type: Research Article

Abstract: This paper is devoted to the mathematical analysis of some algorithms for the computation of the outgoing solution of the Helmholtz equation in an exterior domain. In a first approximation an artificial boundary with absorbing boundary condition is inserted. One then computes the periodic response in this bounded dissipative setting to a periodic forcing term. The response is characterized as the unique minimum of a convex functional. The functional is computed from solution of the time dependent problem in the artificially bounded domain. One such algorithm is due to Glowinski who proposed the functional J2 described below, it has …been implemented by Bristeau et al. [5]. We propose a different functional J1 which is unconditionally coercive while the coerciveness of J2 depends in a subtle way on the geometry of the domain. It is coercive for non-trapping obstacles. The coerciveness property is essential for the convergence of the numerical method. Show more

DOI: 10.3233/ASY-1994-9201

Citation: Asymptotic Analysis, vol. 9, no. 2, pp. 101-117, 1994

Authors: Miara, B.

Article Type: Research Article

Abstract: In the first part of this study, where the linear case was considered, the appropriate scalings of the components of the displacement and the appropriate assumptions on the data (Lamé constants and applied forces) that are an essential step for deriving Kirchhoff-Love plate models by an asymptotic analysis, were justified up to a multiplicative factor. The study of the nonlinear case undertaken here, provides a full justification of this asymptotic analysis by “freezing” this dangling factor.

DOI: 10.3233/ASY-1994-9202

Citation: Asymptotic Analysis, vol. 9, no. 2, pp. 119-134, 1994

Authors: Amar, Micol | Cellina, Arrigo

Article Type: Research Article

Abstract: In this paper, we consider a sequence of integral functionals Fn :X→(−∞,+∞], where X is the set of those functions u belonging to W1,p (0, T), p > 1, satisfying: u(0) = A, u(T) = B. For every n∈N,Fn is represented by the sum of two integrands, where the first one is T-periodic in time and non-convex with respect to u′ and the second one depends only on u. We give a necessary and sufficient condition in order to obtain the existence of an integral functional F∞ :X→(−∞,+∞] such that, for every minimizing sequence (un ) converging to …u∞ , the lower limit of the corresponding sequence Fn (un ) coincides with F∞ (u∞ ). The integrand function in F∞ does not depend on time and, in general, it is non-convex with respect to u′. Show more

DOI: 10.3233/ASY-1994-9203

Citation: Asymptotic Analysis, vol. 9, no. 2, pp. 135-148, 1994

Authors: Mascolo, E. | Migliaccio, L.

Article Type: Research Article

Abstract: We study the asymptotic behaviour of the minimizers of the variational problem: min {∫a −a [ε2 (u′)2 (x)+W(u(x))]dx:u∈L1 (−a,a),u≥0,∫a −a u(x)dx=m,u(−a)=u(a)=c}, where m/2a∈(α,β) and W is a non-negative, continuous and non-convex function, with W(u) = 0 iff u∈{α,β}. We prove that the presence of the necking is related to the value c; more precisely, we have a “neck” if c<α and a “buldge” if c>β.

DOI: 10.3233/ASY-1994-9204

Citation: Asymptotic Analysis, vol. 9, no. 2, pp. 149-161, 1994

Authors: Ozawa, T.

Article Type: Research Article

Abstract: Asymptotic expansions of solutions of the wave equations in even dimensional spaces are obtained with the initial data of non-compact support. A relationship is proved between the vanishing order at the origin of the Fourier transform of the data and the decay rate of the corresponding solutions in semi-infinite cylinders or along rays inside the forward light cone.

DOI: 10.3233/ASY-1994-9205

Citation: Asymptotic Analysis, vol. 9, no. 2, pp. 163-176, 1994