# A computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem

Shuting Cai, Yoshitaka Watanabe

3 引用 (Scopus)

### 抄録

We propose a computer-assisted method for excluding eigenvalues of an elliptic operator linearized at a solution of a nonlinear problem. The method works in both the one-dimensional and the two-dimensional case. We begin by finding an approximate solution to a nonlinear problem, and we then enclose the solution by using Nakao’s numerical verification method. Instead of considering directly the eigenvalues for the elliptic operator linearized at the verified solution, we linearize the operator at the approximate solution. We present a theorem that allows us to determine under which conditions and in which disks there will be no eigenvalues. Thus, if any of those disks are contained in the enclosed area, we can exclude those eigenvalues. Next, we construct various computable criteria that allow us to use a computer program to find these disks. Finally, we use our results to determine which eigenvalues to exclude for the operator linearized at the verified solution. We present some verified results.

元の言語 英語 263-294 32 Japan Journal of Industrial and Applied Mathematics 32 1 https://doi.org/10.1007/s13160-015-0167-7 出版済み - 1 1 2015

### Fingerprint

Elliptic Operator
Nonlinear Problem
Eigenvalue
Approximate Solution
Computer program listings
Numerical Verification
Operator
Theorem

### All Science Journal Classification (ASJC) codes

• Engineering(all)
• Applied Mathematics

### これを引用

：: Japan Journal of Industrial and Applied Mathematics, 巻 32, 番号 1, 01.01.2015, p. 263-294.

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