Asymptotic Analysis - Volume 19, issue 3-4

Authors: Anders, I.

Article Type: Research Article

Abstract: We prove the existence of non‐decaying real solutions of the (2+1) ‐dimensional Gardner equation, vanishing when x\to +\infty . We give asymptotic formulae for the solutions in terms of infinite series of asymptotic solitons with curved lines of constant phase and varying amplitude for \vert t\vert \to\infty .

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 185-207, 1999

Authors: Kerler, Charlotte

Article Type: Research Article

Abstract: We consider self‐adjoint, strictly elliptic, short‐range perturbations of the Laplacian with variable coefficients, that operate on square integrable functions defined in an exterior domain. According to the limiting absorption principle, the resolvent is extended to the positive real line, which is the continuous spectrum of these operators. In suitable weighted spaces, where the weight also depends on the grade of differentiation, we show that the extended resolvent is strongly differentiable w.r.t. the spectral parameter. Moreover, we obtain estimates for the derivatives of the extended resolvent in these weighted spaces, which are locally uniform in the spectral parameter.

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 209-232, 1999

Authors: Goldstein, Gisèle Ruiz | Goldstein, Jerome A. | Obrecht, Enrico

Article Type: Research Article

Abstract: Scattering theory tells how solutions of one abstract Schrödinger equation (of the form {\rm i}({\rm d}u/{\rm d}t)=Hu with H=H^*) are asymptotic to solutions of another (in principle simpler) abstract Schrödinger equation. We extend this theory to inhomogeneous problems of the form {\rm i}({\rm d}u)/({\rm d}t)=Hu+h(t) , with special emphasis on factored equations of the form \prod_{j=1}^N (({\rm d}/{\rm d}t)-{\rm i}A_j)u(t) = h(t), where A_1,\ldots, A_N are commuting selfadjoint operators. As a special case, corresponding to N=4 and two‐space scattering, we conclude that every solution u(\cdot, t) of …the inhomogeneous elastic wave equation in the exterior of a bounded star shaped obstacle is of the form u=v+w+z, where v(\cdot, t) solves the free (homogeneous) elastic wave equation with no obstacle, w(\cdot,t) is determined by the (rather general) inhomogeneity, and z(\cdot,t)={\rm o}(1) as t\to \pm \infty. Some of the results are presented in a more general Banach space context. Show more

Keywords: Scattering, d’Alembert’s formula, factored equations, elastic waves, Duhamel’s principle, unitary groups, (C_0) semigroups, asymptotics

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 233-252, 1999

Authors: Popov, Georgi | Vodev, Georgi

Article Type: Research Article

Abstract: We study the resonances associated to the transmission problem for a strictly convex obstacle provided that the speed of propagation of the waves in the interior of the obstacle is strictly greater than the speed in the exterior. We prove that there are no resonances in a region of the form {\rm Im}\,z\leqslant C_1|z|^{-1},\ |{\rm Re}\,z|\geqslant C_2>0 . Using this we obtain some uniform estimates on the decay of the local energy.

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 253-265, 1999

Authors: Feireisl, Eduard | Laurençot, Philippe

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 267-288, 1999

Authors: Tate, Tatsuya

Article Type: Research Article

Abstract: Our purpose of this paper is to estimate the rate of the decay of the off‐diagonal asymptotics in Sunada (preprint) in case where the Hamilton flow is of Anosov type. For the sake of this, we use the central limit theorem for transitive Anosov flows (Sinai, Soviet Math. Dokl. 1 (1960), 983–987; Ratner, Israel J. Math. 16 (1973), 181–197; Zelditch, Comm. Math. Phys. 160 (1994), 81–92). Also it is shown that if the Hamilton flow has homogeneous Lebesgue spectrum, then the measure {\rm d}m_{A} associated with a pseudodifferential operator A , which is introduced by Zelditch (J. …Funct. Anal. 140 (1996), 68–86), is absolutely continuous with respect to Lebesgue measure. Show more

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 289-296, 1999

Authors: Anné, Colette

Article Type: Research Article

Abstract: We consider the equation of linear elasticity on a general Riemannian manifold with boundary, and prove a formula relating the counting functions of the Neumann and the Dirichlet problem to the counting function of the Dirichlet‐to‐Neumann operator. Namely, the difference of the two counting functions at \alpha equals the number of negative eigenvalues of the Dirichlet‐to‐Neumann operator related to the resolvent at \alpha . We then apply this formula to bounded domains of Riemannian symmetric spaces of non‐compact type in the homogeneous case of elasticity (i.e., when the Lamé functions \lambda , \mu …are constant). The conclusion is that the difference of the two counting functions is greater or equal to 1 under one of the following hypothesis: either the rank of the symmetric space is greater or equal to 2, or the rank is 1 but the dimension of the nilpotent part is smaller than {8\mu}/({\lambda+2\mu}) . The Euclidean space is an example of the first case, but even in that situation the conclusion we draw is new. Show more

Keywords: Elasticity, boundary conditions, spectrum, symmetric spaces

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 297-316, 1999

Authors: Chipot, M. | Kiss, L.

Article Type: Research Article

Citation: Asymptotic Analysis, vol. 19, no. 3‐4, pp. 317-341, 1999