### Abstract

We consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation. This problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems. We formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation. We also consider the indices of eigenvalues, especially the first eigenvalue of such a problem. In these enclosing procedures, the finite-dimensional verified computations for linear and nonlinear system of equations play an essential role. A numerical example is presented.

Original language | English |
---|---|

Pages (from-to) | 81-106 |

Number of pages | 26 |

Journal | Linear Algebra and Its Applications |

Volume | 324 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - 2001 Feb 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Eigenvalue enclosing
- Linearized eigenvalue problem
- Nonlinear elliptic PDE.

### ASJC Scopus subject areas

- Algebra and Number Theory
- Numerical Analysis
- Geometry and Topology
- Discrete Mathematics and Combinatorics

### Cite this

*Linear Algebra and Its Applications*,

*324*(1-3), 81-106. https://doi.org/10.1016/S0024-3795(00)00127-0

**An enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems.** / Nagatou, K.; Nakao, M. T.

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 324, no. 1-3, pp. 81-106. https://doi.org/10.1016/S0024-3795(00)00127-0

}

TY - JOUR

T1 - An enclosure method of eigenvalues for the elliptic operator linearized at an exact solution of nonlinear problems

AU - Nagatou, K.

AU - Nakao, M. T.

PY - 2001/2/15

Y1 - 2001/2/15

N2 - We consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation. This problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems. We formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation. We also consider the indices of eigenvalues, especially the first eigenvalue of such a problem. In these enclosing procedures, the finite-dimensional verified computations for linear and nonlinear system of equations play an essential role. A numerical example is presented.

AB - We consider eigenvalue enclosing of the elliptic operator which is linearized at an exact solution of certain nonlinear elliptic equation. This problem is important in the mathematically rigorous analysis of the stability or bifurcation of some solutions for nonlinear problems. We formulate such a kind of eigenvalue problem as the nonlinear system which contains both linearized eigenvalue problem and the original nonlinear equation. We also consider the indices of eigenvalues, especially the first eigenvalue of such a problem. In these enclosing procedures, the finite-dimensional verified computations for linear and nonlinear system of equations play an essential role. A numerical example is presented.

KW - Eigenvalue enclosing

KW - Linearized eigenvalue problem

KW - Nonlinear elliptic PDE.

UR - http://www.scopus.com/inward/record.url?scp=0035632268&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035632268&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(00)00127-0

DO - 10.1016/S0024-3795(00)00127-0

M3 - Article

AN - SCOPUS:0035632268

VL - 324

SP - 81

EP - 106

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -