### 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.

Original language | English |
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Pages (from-to) | 103-123 |

Number of pages | 21 |

Journal | Asymptotic Analysis |

Volume | 65 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

### Fingerprint

### Keywords

- Asymptotic behaviour
- Elliptic eigenvalue problems

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Asymptotic Analysis*,

*65*(1-2), 103-123. https://doi.org/10.3233/ASY-2009-0951

**Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type.** / Jimbo, Shuichi; Kimura, Masato; Notsu, Hirofumi.

Research output: Contribution to journal › Article

*Asymptotic Analysis*, vol. 65, no. 1-2, pp. 103-123. https://doi.org/10.3233/ASY-2009-0951

}

TY - JOUR

T1 - Exponential decay phenomenon of the principal eigenvalue of an elliptic operator with a large drift term of gradient type

AU - Jimbo, Shuichi

AU - Kimura, Masato

AU - Notsu, Hirofumi

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Asymptotic behaviour

KW - Elliptic eigenvalue problems

UR - http://www.scopus.com/inward/record.url?scp=72649097669&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=72649097669&partnerID=8YFLogxK

U2 - 10.3233/ASY-2009-0951

DO - 10.3233/ASY-2009-0951

M3 - Article

AN - SCOPUS:72649097669

VL - 65

SP - 103

EP - 123

JO - Asymptotic Analysis

JF - Asymptotic Analysis

SN - 0921-7134

IS - 1-2

ER -