TY - JOUR
T1 - Inverse norm estimation of perturbed Laplace operators and corresponding eigenvalue problems
AU - Sekine, Kouta
AU - Tanaka, Kazuaki
AU - Oishi, Shin'ichi
N1 - Funding Information:
This work is supported by JST CREST Grant Number JPMJCR14D4 , and MEXT as “Exploratory Issue on Post-K computer” (Development of verified numerical computations and super high-performance computing environment for extreme researches) Grant Number hp160255 . The first author (K.S.) is supported by JSPS KAKENHI Grant Number 16K17651 . The second author (K.T.) is supported by JSPS KAKENHI Grant Number 19K14601 . We thank the editors and reviewers for providing useful comments that helped us to improve the content of this manuscript.
Publisher Copyright:
© 2021 The Authors
PY - 2022/1/15
Y1 - 2022/1/15
N2 - In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.
AB - In numerical existence proofs for solutions of the semi-linear elliptic system, evaluating the norm of the inverse of a perturbed Laplace operator plays an important role. We reveal an eigenvalue problem to design a method for verifying the invertibility of the operator and evaluating the norm of its inverse based on Liu's method and the Temple-Lehmann-Goerisch method. We apply the inverse-norm's estimation to the Dirichlet boundary value problem of the Lotka-Volterra system with diffusion terms and confirm the efficacy of our method.
KW - Computer-assisted proofs
KW - Eigenvalue evaluation
KW - Norm of inverse operators
KW - Rigorous numerical computations
KW - System of partial differential equations
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U2 - 10.1016/j.camwa.2021.12.002
DO - 10.1016/j.camwa.2021.12.002
M3 - Article
AN - SCOPUS:85120953550
SN - 0898-1221
VL - 106
SP - 18
EP - 26
JO - Computers and Mathematics with Applications
JF - Computers and Mathematics with Applications
ER -