Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type

Article type: Research Article

Authors: Jimbo, Shuichi | Kimura, Masato^; | Notsu, Hirofumi

Affiliations: Department of Mathematics, Hokkaido University, Sapporo, Japan. E-mail: jimbo@math.sci.hokudai.ac.jp | Faculty of Mathematics, Kyushu University, Fukuoka, Japan. E-mails: {masato, notsu}@math.kyushu-u.ac.jp

Note: [] Corresponding author: Masato Kimura, Faculty of Mathematics, Kyushu University, Hakozaki 6-10-1, Higashi-ku, Fukuoka, 812-8581, Japan. Tel.: +81 92 642 4348; Fax: +81 92 642 2779; E-mail: masato@math.kyushu-u.ac.jp.

Abstract: We study an asymptotic behaviour of the principal eigenvalue for an elliptic operator with large advection which is given by a gradient of a potential function. It is shown that the principal eigenvalue decays exponentially under the velocity potential well condition as the parameter tends to infinity. We reveal that the depth of the potential well plays an important role in the estimate. Particularly, in one-dimensional case, we give a much more elaborate characterization for the eigenvalue. Some numerical examples are also shown.

Keywords: elliptic eigenvalue problems, asymptotic behaviour

DOI: 10.3233/ASY-2009-0951

Journal: Asymptotic Analysis, vol. 65, no. 1-2, pp. 103-123, 2009