Asymptotic Analysis - Volume 8, issue 4

Authors: Alama, S. | Avellaneda, M. | Deift, P.A. | Hempel, R.

Article Type: Research Article

Abstract: We consider uniformly elliptic divergence type operators A=−Σ∂j aij (x)∂i with bounded, Lipschitz continuous coefficients, acting in the Hilbert space L2 (Rν ). It is easy to see that such an operator cannot have discrete eigenvalues below the infimum of the essential spectrum. In order to produce eigenvalues with exponentially decaying eigenfunctions we study the family of operators A+λB=−Σ∂j (aij (x)+λbij (x))∂i , λ≥0, where A is supposed to have a spectral gap, while (bij )≥0 and bij (x)→0, as x→∞. One of our main results assures that discrete eigenvalues of A+λB move into the gap, as λ …increases, if the support of the matrix function (bij ) is large enough. In addition, we analyze the connection between decay properties of the coefficient matrix (bij ) and the asymptotics of the associated eigenvalue counting function; these results are modeled on our earlier work in the Schrödinger case. Show more

DOI: 10.3233/ASY-1994-8401

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 311-344, 1994

Authors: Berger, Günter

Article Type: Research Article

Abstract: Spectral properties of strongly elliptic operators of second order on bounded domains with infinitely many hooks and Neumann boundary conditions are considered. In the case of a pure point spectrum asymptotic formulae with remainder estimates for the counting function N(λ) of the eigenvalues are proved. These formulae yield the conclusion that the asymptotic eigenvalue distributions coincide with the classical formula of Weyl.

DOI: 10.3233/ASY-1994-8402

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 345-362, 1994

Authors: Maier, Stanislaus

Article Type: Research Article

Abstract: We study solutions u=u(t) of an initial value problem for u″+(n−1)/t u′+f(u)=0, which have the additional property that the limit of u as t approaches infinity does not exist. Besides some examples, we give necessary conditions on the non-linearity f for the existence of such (non-convergent) solutions. One corollary of these investigations will be that, if f(u)u>0 for small |u|≠0 then every solution u(·;p) (with |p| small enough) of the initial value problem (1.1), stated below, converges to zero as t→∞.

DOI: 10.3233/ASY-1994-8403

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 363-377, 1994

Authors: Goh, William M.Y. | Wimp, Jet

Article Type: Research Article

Abstract: In this paper we introduce a singular function which reflects some of the behavior of both the classical Cantor singular function and of the Riesz–Nagy class of singular functions. We determine the asymptotic behavior of the corresponding moment sequence. The behavior of the moments is similar to the behavior of the moments of two singular functions we have considered previously.

DOI: 10.3233/ASY-1994-8404

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 378-392, 1994

Authors: Miersemann, Erich

Article Type: Research Article

Abstract: The asymptotic correctness is shown of a formal expansion given by Laplace in 1806 of the rise height of a fluid in a circular capillary tube. The proof is completely based on the comparison principle of Concus and Finn. The first two non-constant terms in the expansion are calculated. It is of special interest that the expansion is uniform with respect to the boundary contact angle although the governing quasilinear elliptic equation becomes singular on the boundary if the contact angle tends to zero.

DOI: 10.3233/ASY-1994-8405

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 393-403, 1994

Article Type: Other

Citation: Asymptotic Analysis, vol. 8, no. 4, pp. 405-405, 1994